Properties of Circles

Size: px
Start display at page:

Download "Properties of Circles"

Transcription

1 roperties of ircles Use roperties of Tangents 10.2 ind rc Measures 10.3 pply roperties of hords 10.4 Use Inscribed ngles and olygons 10.5 pply Other ngle elationships in ircles 10.6 ind egment Lengths in ircles 10.7 Write and Graph quations of ircles efore In previous chapters, you learned the following skills, which you ll use in hapter 10: classifying triangles, finding angle measures, and solving equations. rerequisite kills VOULY HK opy and complete the statement. 1. Two similar triangles have congruent corresponding angles and? corresponding sides. 2. Two angles whose sides form two pairs of opposite rays are called?. 3. The? of an angle is all of the points between the sides of the angle. KILL N LG HK Use the onverse of the ythagorean Theorem to classify the triangle. (eview p. 441 for 10.1.) , 0.8, , 12, , 2, 2.5 ind the value of the variable. (eview pp. 24, 35 for 10.2, 10.4.) (5 1 40)8 5 8 (6 2 8)8 (8 2 2)8 (2 1 2)

2 Now In hapter 10, you will apply the big ideas listed below and reviewed in the hapter ummary on page 707. You will also use the key vocabulary listed below. ig Ideas 1 Using properties of segments that intersect circles 2 pplying angle relationships in circles 3 Using circles in the coordinate plane KY VOULY circle, p. 651 center, radius, diameter chord, p. 651 secant, p. 651 tangent, p. 651 central angle, p. 659 minor arc, p. 659 major arc, p. 659 semicircle, p. 659 congruent circles, p. 660 congruent arcs, p. 660 inscribed angle, p. 672 intercepted arc, p. 672 standard equation of a circle, p. 699 Why? ircles can be used to model a wide variety of natural phenomena. You can use properties of circles to investigate the Northern Lights. Geometry The animation illustrated below for ample 4 on page 682 helps you answer this question: rom what part of arth are the Northern Lights visible? Your goal is to determine from what part of arth you can see the Northern Lights. To begin, complete a justification of the statement that >. Geometry at classzone.com Other animations for hapte r 10: page s 655, 661, 671, 691, and

3 10.1 plore Tangent egments MTIL compass ruler TIVITY Investigating Geometry Use before Lesson 10.1 QUTION How are the lengths of tangent segments related? line can intersect a circle at 0, 1, or 2 points. If a line is in the plane of a circle and intersects the circle at 1 point, the line is a tangent. XLO raw tangents to a circle T 1 T 2 T 3 raw a circle Use a compass to draw a circle. Label the center. raw tangents raw lines ] and ] so that they intersect ( only at and, respectively. These lines are called tangents. Measure segments } and } are called tangent segments. Measure and compare the lengths of the tangent segments. W ONLUION Use your observations to complete these eercises 1. epeat teps 1 3 with three different circles. 2. Use your results from ercise 1 to make a conjecture about the lengths of tangent segments that have a common endpoint. 3. In the diagram, L, Q, N, and are points of tangency. Use your conjecture from ercise 2 to find LQ and N if LM 5 7 and M L N M 4. In the diagram below,,,, and are points of tangency. Use your conjecture from ercise 2 to eplain why } > }. 650 hapter 10 roperties of ircles

4 Use roperties 10.1 of Tangents efore You found the circumference and area of circles. Now You will use properties of a tangent to a circle. Why? o you can find the range of a G satellite, as in. 37. Key Vocabulary circle center, radius, diameter chord secant tangent circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. circle with center is called circle and can be written (. segment whose endpoints are the center and any point on the circle is a radius. chord is a segment whose endpoints are on a circle. diameter is a chord that contains the center of the circle. chord radius diameter center secant is a line that intersects a circle in two points. tangent is a line in the plane of a circle that intersects the circle in eactly one point, the point of tangency. The tangent ray ] and the tangent segment } are also called tangents. secant point of tangency tangent XML 1 Identify special segments and lines Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of (. a. } b. } c. ] d. ] G olution a. } is a radius because is the center and is a point on the circle. b. } is a diameter because it is a chord that contains the center. ] c. is a tangent ray because it is contained in a line that intersects the circle at only one point. ] d. is a secant because it is a line that intersects the circle in two points. GUI TI for ample 1 1. In ample 1, what word best describes } G? }? 2. In ample 1, name a tangent and a tangent segment Use roperties of Tangents 651

5 VOULY The plural of radius is radii. ll radii of a circle are congruent. IU N IMT The words radius and diameter are used for lengths as well as segments. or a given circle, think of a radius and a diameter as segments and the radius and the diameter as lengths. XML 2 ind lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. adius of ( y b. iameter of ( c. adius of ( d. iameter of ( 1 olution 1 a. The radius of ( is 3 units. b. The diameter of ( is 6 units. c. The radius of ( is 2 units. d. The diameter of ( is 4 units. GUI TI for ample 2 3. Use the diagram in ample 2 to find the radius and diameter of ( and (. OLN IL Two circles can intersect in two points, one point, or no points. oplanar circles that intersect in one point are called tangent circles. oplanar circles that have a common center are called concentric. concentric circles 2 points of intersection 1 point of intersection (tangent circles) no points of intersection VOULY line that intersects a circle in eactly one point is said to be tangent to the circle. OMMON TNGNT line, ray, or segment that is tangent to two coplanar circles is called a common tangent. common tangents 652 hapter 10 roperties of ircles

6 XML 3 raw common tangents Tell how many common tangents the circles have and draw them. a. b. c. olution a. 4 common tangents b. 3 common tangents c. 2 common tangents GUI TI for ample 3 Tell how many common tangents the circles have and draw them THOM or Your Notebook THOM 10.1 In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. m roof: s , p. 658 Line m is tangent to (Q if and only if m } Q. XML 4 Verify a tangent to a circle In the diagram, } T is a radius of (. Is } T tangent to (? 35 T olution Use the onverse of the ythagorean Theorem. ecause , nt is a right triangle and } T } T. o, } T is perpendicular to a radius of ( at its endpoint on (. y Theorem 10.1, } T is tangent to ( Use roperties of Tangents 653

7 XML 5 ind the radius of a circle In the diagram, is a point of tangency. ind the radius r of (. olution You know from Theorem 10.1 that } },son is a right triangle. You can use the ythagorean Theorem ythagorean Theorem (r 1 50) 2 5 r ubstitute. r r r Multiply. 100r ubtract from each side. r 5 39 ft ivide each side by ft 80 ft r r THOM THOM 10.2 Tangent segments from a common eternal point are congruent. roof:. 41, p. 658 or Your Notebook T If } and } T are tangent segments, then } > } T. XML 6 ind the radius of a circle } is tangent to ( at and } T is tangent to ( at T. ind the value of. olution T 5 T Tangent segments from the same point are > ubstitute. 85 olve for. GUI TI for amples 4, 5, and 6 7. Is } tangent to (? 8. } T is tangent to (Q. 9. ind the value(s) ind the value of r. of r r T hapter 10 roperties of ircles

8 10.1 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 7, 19, and 37 5 TNIZ TT TI s. 2, 29, 33, and VOULY opy and complete: The points and are on (. If is a point on }, then } is a?. 2. WITING plain how you can determine from the contet whether the words radius and diameter are referring to a segment or a length. XML 1 on p. 651 for s MTHING TM Match the notation with the term that best describes it. 3.. enter ] 4. H. adius 5. }. hord ] 6.. iameter ] 7.. ecant 8. G. Tangent 9. } G. oint of tangency 10. } H. ommon tangent G H at classzone.com 11. O NLYI escribe and correct the error in the statement about the diagram. 6 9 The length of secant } is 6. XML 2 and 3 on pp for s OOINT GOMTY Use the diagram at the right. 12. What are the radius and diameter of (? 13. What are the radius and diameter of (? 14. opy the circles. Then draw all the common tangents of the two circles y WING TNGNT opy the diagram. Tell how many common tangents the circles have and draw them Use roperties of Tangents 655

9 XML 4 on p. 653 for s XML 5 and 6 on p. 654 for s TMINING TNGNY etermine whether } is tangent to (. plain LG ind the value(s) of the variable. In ercises 24 26, and are points of tangency. 21. r r r 18 r r r OMMON TNGNT common internal tangent intersects the segment that joins the centers of two circles. common eternal tangent does not intersect the segment that joins the centers of the two circles. etermine whether the common tangents shown are internal or eternal MULTIL HOI In the diagram, ( and (Q are tangent circles. } is a common tangent. ind. 22Ï } Ï } ONING In the diagram, ] is tangent to (Q and (. plain why } > } > } even though the radius of (Q is not equal to the radius of (. 31. TNGNT LIN When will two lines tangent to the same circle not intersect? Use Theorem 10.1 to eplain your answer WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

10 32. NGL ITO In the diagram at right, and are points of tangency on (. plain how you know that ] bisects. (Hint: Use Theorem 5.6, page 310.) 33. HOT ON or any point outside of a circle, is there ever only one tangent to the circle that passes through the point? re there ever more than two such tangents? plain your reasoning. 34. HLLNG In the diagram at the right, , 5 8, and all three segments are tangent to (. What is the radius of (? OLM OLVING IYL On modern bicycles, rear wheels usually have tangential spokes. Occasionally, front wheels have radial spokes. Use the definitions of tangent and radius to determine if the wheel shown has tangential spokes or radial spokes XML 4 on p. 653 for GLOL OITIONING YTM (G) G satellites orbit about 11,000 miles above arth. The mean radius of arth is about 3959 miles. ecause G signals cannot travel through arth, a satellite can transmit signals only as far as points and from point, as shown. ind and to the nearest mile. 38. HOT ON In the diagram, } is a common internal tangent (see ercises 27 28) to ( and (. Use similar triangles to eplain why } 5 } Use roperties of Tangents 657

11 39. OVING THOM 10.1 Use parts (a) (c) to prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. GIVN c Line m is tangent to (Q at. OV c m } Q m a. ssume m is not perpendicular to } Q. Then the perpendicular segment from Q to m intersects m at some other point. ecause m is a tangent, cannot be inside (Q. ompare the length Q to Q. b. ecause } Q is the perpendicular segment from Q to m, } Q is the shortest segment from Q to m. Now compare Q to Q. c. Use your results from parts (a) and (b) to complete the indirect proof. 40. OVING THOM 10.1 Write an indirect proof that if a line is perpendicular to a radius at its endpoint, the line is a tangent. GIVN c m } Q OV c Line m is tangent to (Q. m 41. OVING THOM 10.2 Write a proof that tangent segments from a common eternal point are congruent. GIVN c } and } T are tangent to (. OV c } > } T lan for roof Use the Hypotenuse Leg ongruence Theorem to show that n > nt. T 42. HLLNG oint is located at the origin. Line l is tangent to ( at (24, 3). Use the diagram at the right to complete the problem. a. ind the slope of line l. (24, 3) y l b. Write the equation for l. c. ind the radius of (. d. ind the distance from l to ( along the y-ais. MIX VIW VIW repare for Lesson 10.2 in is in the interior of. If m and m 5 708, find m. (p. 24) ind the values of and y. (p. 154) y y8 46. (4y 2 7)8 (2 1 3) triangle has sides of lengths 8 and 13. Use an inequality to describe the possible length of the third side. What if two sides have lengths 4 and 11? (p. 328) 658 XT TI for Lesson 10.1, p. 914 ONLIN QUIZ at classzone.com

12 10.2 ind rc Measures efore You found angle measures. Now You will use angle measures to find arc measures. Why? o you can describe the arc made by a bridge, as in. 22. Key Vocabulary central angle minor arc major arc semicircle measure minor arc, major arc congruent circles congruent arcs central angle of a circle is an angle whose verte is the center of the circle. In the diagram, is a central angle of (. If m is less than 1808, then the points on ( that lie in the interior of form a minor arc with endpoints and. The points on ( that do not lie on minor arc form a major arc with endpoints and. semicircle is an arc with endpoints that are the endpoints of a diameter. minor NMING Minor arcs are named by their endpoints. The minor arc associated with is named. Major arcs and semicircles are named by their endpoints and a point on the arc. The major arc associated with can be named. major arc $ KY ONT or Your Notebook Measuring rcs The measure of a minor arc is the measure of its central angle. The epression m is read as the measure of arc. The measure of the entire circle is The measure of a major arc is the difference between 3608 and the measure of the related minor arc. The measure of a semicircle is m m XML 1 ind measures of arcs ind the measure of each arc of (, where } T is a diameter. a. b. T c. T olution T 1108 a. is a minor arc, so m 5 m b. T is a major arc, so m T c. } T is a diameter, so T is a semicircle, and m T ind rc Measures 659

13 JNT Two arcs of the same circle are adjacent if they have a common endpoint. You can add the measures of two adjacent arcs. OTULT OTULT 23 rc ddition ostulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. or Your Notebook m 5 m 1 m XML 2 ind measures of arcs UVY recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. ind the indicated arc measures. a. m b. m c. m d. m Whom Would You ather Meet? Musician thlete Inventor 798 Other ctor MU The measure of a minor arc is less than The measure of a major arc is greater than olution a. m 5 m 1 m b. m 5 m 1 m c. m m d. m m GUI TI for amples 1 and 2 Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. 1. TQ 2. QT 3. TQ 4. Q 5. T 6. T T ONGUNT IL N Two circles are congruent circles if they have the same radius. Two arcs are congruent arcs if they have the same measure and they are arcs of the same circle or of congruent circles. If ( is congruent to (, then you can write ( > (. 660 hapter 10 roperties of ircles

14 XML 3 Identify congruent arcs Tell whether the red arcs are congruent. plain why or why not. a. b. T c U V 958 X Y 958 Z olution a. > because they are in the same circle and m 5 m. b. and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent. c. VX > YZ because they are in congruent circles and mvx 5 myz. at classzone.com GUI TI for ample 3 Tell whether the red arcs are congruent. plain why or why not M N XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 5, 13, and 23 5 TNIZ TT TI s. 2, 11, 17, 18, and VOULY opy and complete: If and are congruent central angles of (, then and are?. 2. WITING What do you need to know about two circles to show that they are congruent? plain. XML 1 and 2 on pp for s MUING } and } are diameters of (. etermine whether the arc is a minor arc, a major arc, or a semicircle of (. Then find the measure of the arc ind rc Measures 661

15 11. MULTIL HOI In the diagram, } Q is a diameter of (. Which arc represents a semicircle? Q Q QT QT T XML 3 on p. 661 for s ONGUNT Tell whether the red arcs are congruent. plain why or why not L 858 M 14. V W X N Y Z 15. O NLYI plain what is wrong with the statement. You cannot tell if ( > ( because the radii are not given. 16. Two diameters of ( are } and }.Ifm 5 208, find m and m. 17. MULTIL HOI ( has a radius of 3 and has a measure of 908. What is the length of }? 3Ï } 2 3Ï } HOT ON On (, m , m G , and m G If H is on ( so that m GH , eplain why H must be on. 19. ONING In (, m 5 608, m 5 258, m 5 708, and m ind two possible values for m. 20. HLLNG In the diagram shown, } Q }, }Q is tangent to (, and mv What is mu? U V 21. HLLNG In the coordinate plane shown, is at the origin. ind the following arc measures on (. a. m y (3, 4) (4, 3) b. m c. m (5, 0) WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

16 OLM OLVING XML 1 on p. 659 for IG The deck of a bascule bridge creates an arc when it is moved from the closed position to the open position. ind the measure of the arc. 23. T On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle? 24. XTN ON surveillance camera is mounted on a corner of a building. It rotates clockwise and counterclockwise continuously between Wall and Wall at a rate of 108 per minute. a. What is the measure of the arc surveyed by the camera? b. How long does it take the camera to survey the entire area once? c. If the camera is at an angle of 858 from Wall while rotating counterclockwise, how long will it take for the camera to return to that same position? d. The camera is rotating counterclockwise and is 508 from Wall. ind the location of the camera after 15 minutes. 25. HLLNG clock with hour and minute hands is set to 1:00.M. a. fter 20 minutes, what will be the measure of the minor arc formed by the hour and minute hands? b. t what time before 2:00.M., to the nearest minute, will the hour and minute hands form a diameter? MIX VIW VIW repare for Lesson 10.3 in s etermine if the lines with the given equations are parallel. (p. 180) 26. y , y 5 5(1 2 ) 27. 2y , y Trace nxyz and point. raw a counterclockwise rotation of nxyz 1458 about. (p. 598) X Z Y ind the product. (p. 641) 29. ( 1 2)( 1 3) 30. (2y 2 5)(y 1 7) 31. ( 1 6)( 2 6) 32. (z 2 3) (3 1 7)(5 1 4) 34. (z 2 1)(z 2 4) XT TI for Lesson 10.2, p. 914 ONLIN QUIZ at classzone.com 663

17 10.3 pply roperties of hords efore You used relationships of central angles and arcs in a circle. Now You will use relationships of arcs and chords in a circle. Why? o you can design a logo for a company, as in. 25. Key Vocabulary chord, p. 651 arc, p. 659 semicircle, p. 659 ecall that a chord is a segment with endpoints on a circle. ecause its endpoints lie on the circle, any chord divides the circle into two arcs. diameter divides a circle into two semicircles. ny other chord divides a circle into a minor arc and a major arc. semicircle diameter semicircle major arc chord minor arc THOM THOM 10.3 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. roof: s , p. 669 or Your Notebook > if and only if } > }. XML 1 Use congruent chords to find an arc measure In the diagram, ( > (Q, } G > } JK, and mjk ind m G. olution ecause } G and } JK are congruent chords in congruent circles, the corresponding minor arcs G and JK are congruent. c o, m G 5 m JK G 808 J K GUI TI for ample 1 Use the diagram of (. 1. If m , find m. 2. If m , find m hapter 10 roperties of ircles

18 ITING If XY > YZ, then the point Y, and any line, segment, or ray that contains Y, bisects XYZ. X } Y bisects XYZ. Y Z THOM or Your Notebook THOM 10.4 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If } Q is a perpendicular bisector of } T, then } Q is a diameter of the circle. roof:. 31, p. 670 T THOM 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If } G is a diameter and } G }, then } H > } H and G > G. roof:. 32, p. 670 H G XML 2 Use perpendicular bisectors GNING Three bushes are arranged in a garden as shown. Where should you place a sprinkler so that it is the same distance from each bush? olution T 1 T 2 T 3 sprinkler Label the bushes,, and, as shown. raw segments } and }. raw the perpendicular bisectors of } and }. y Theorem 10.4, these are diameters of the circle containing,, and. ind the point where these bisectors intersect. This is the center of the circle through,, and, and so it is equidistant from each point pply roperties of hords 665

19 XML 3 Use a diameter Use the diagram of ( to find the length of }. Tell what theorem you use. olution iameter } is perpendicular to }. o, by Theorem 10.5, } bisects },and 5. Therefore, 5 2( ) 5 2(7) GUI TI for amples 2 and 3 ind the measure of the indicated arc in the diagram (80 2 )8 THOM THOM 10.6 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. roof:. 33, p. 670 or Your Notebook G } > } if and only if 5 G. XML 4 Use Theorem 10.6 In the diagram of (, Q 5 T ind U. olution hords } Q and } T are congruent, so by Theorem 10.6 they are equidisant from. Therefore, U 5 V. U 5 V Use Theorem ubstitute. 5 3 olve for. c o, U (3) U V T 16 GUI TI for ample 4 In the diagram in ample 4, suppose T 5 32, and U 5 V ind the given length. 6. Q 7. QU 8. The radius of ( 666 hapter 10 roperties of ircles

20 10.3 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 7, 9, and 25 5 TNIZ TT TI s. 2, 15, 22, and VOULY escribe what it means to bisect an arc. 2. WITING Two chords of a circle are perpendicular and congruent. oes one of them have to be a diameter? plain your reasoning. XML 1 and 3 on pp. 664, 666 for s. 3 5 INING MU ind the measure of the red arc or chord in ( G 8 J H XML 3 and 4 on p. 666 for s LG ind the value of in (Q. plain your reasoning M L N U T H G ONING In ercises 12 14, what can you conclude about the diagram shown? tate a theorem that justifies your answer H G J 14. N L M 15. MULTIL HOI In the diagram of (, which congruence relation is not necessarily true? } Q > } QN MN > M } NL > } L } N > } L M L N 10.3 pply roperties of hords 667

21 16. O NLYI plain what is 17. O NLYI plain why the wrong with the diagram of (. congruence statement is wrong. 6 6 G 7 7 H > INTIYING IMT etermine whether } is a diameter of the circle. plain your reasoning ONING In the diagram of semicircle Q, } ù } and m plain how you can conclude that n ù n. 22. WITING Theorem 10.4 is nearly the converse of Theorem a. Write the converse of Theorem plain how it is different from Theorem b. opy the diagram of ( and draw auiliary segments } and }. Use congruent triangles to prove the converse of Theorem c. Use the converse of Theorem 10.5 to show that Q 5 Q in the diagram of (. 23. LG In ( below, }, }, 24. HLLNG In ( below, the and all arcs have integer measures. how that must be even. T lengths of the parallel chords are 20, 16, and 12. ind m. Q WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

22 OLM OLVING 25. LOGO IGN The owner of a new company would like the company logo to be a picture of an arrow inscribed in a circle, as shown. or symmetry, she wants to be congruent to. How should } and } be related in order for the logo to be eactly as desired? XML 2 on p. 665 for ON-N MTH In the cross section of the submarine shown, the control panels are parallel and the same length. plain two ways you can find the center of the cross section. OVING THOM 10.3 In ercises 27 and 28, prove Theorem GIVN c } and } are congruent chords. OV c > 28. GIVN c } and } are chords and >. OV c } > } 29. HO LNGTH Make and prove a conjecture about chord lengths. a. ketch a circle with two noncongruent chords. Is the longer chord or the shorter chord closer to the center of the circle? epeat this eperiment several times. b. orm a conjecture related to your eperiment in part (a). c. Use the ythagorean Theorem to prove your conjecture. 30. MULTI-T OLM If a car goes around a turn too quickly, it can leave tracks that form an arc of a circle. y finding the radius of the circle, accident investigators can estimate the speed of the car. a. To find the radius, choose points and on the tire marks. Then find the midpoint of }. Measure }, as shown. ind the radius r of the circle. b. The formula Ï } fr can be used to estimate a car s speed in miles per hours, where f is the coefficient of friction and r is the radius of the circle in feet. The coefficient of friction measures how slippery a road is. If f 5 0.7, estimate the car s speed in part (a) pply roperties of hords 669

23 OVING THOM 10.4 N 10.5 Write proofs. 31. GIVN c } Q is the perpendicular 32. GIVN c } G is a diameter of (L. bisector of } T. G } } OV c } Q is a diameter of (L. OV c } > }, G > G lan for roof Use indirect reasoning. lan for roof raw } L and } L. ssume center L is not on } Q.rove Use congruent triangles to show that nl > ntl, so } L } T. Then } > } and LG > LG. use the erpendicular ostulate. Then show G > G. L T G L 33. OVING THOM 10.6 or Theorem 10.6, prove both cases of the biconditional. Use the diagram shown for the theorem on page HLLNG car is designed so that the rear wheel is only partially visible below the body of the car, as shown. The bottom panel is parallel to the ground. rove that the point where the tire touches the ground bisects. MIX VIW VIW repare for Lesson 10.4 in s The measures of the interior angles of a quadrilateral are 1008, 1408, ( 1 20)8, and (2 1 10)8. ind the value of. (p. 507) Quadrilateral JKLM is a parallelogram. Graph ~JKLM. ecide whether it is best described as a rectangle, a rhombus, or a square. (p. 552) 36. J(23, 5), K(2, 5), L(2, 21), M(23, 21) 37. J(25, 2), K(1, 1), L(2, 25), M(24, 24) QUIZ for Lessons etermine whether } is tangent to (. plain your reasoning. (p. 651) If m G , and m 5 808, find m G and m G. (p. 659) 4. The points,, and are on (, } > },andm What is the measure of? (p. 664) 670 XT TI for Lesson 10.3, p. 914 ONLIN QUIZ at classzone.com

24 TIVITY Investigating Geometry Use before Lesson plore Inscribed ngles MTIL compass straightedge protractor QUTION How are inscribed angles related to central angles? The verte of a central angle is at the center of the circle. The verte of an inscribed angle is on the circle, and its sides form chords of the circle. XLO onstruct inscribed angles of a circle T 1 T 2 T 3 U T U T V V raw a central angle Use a compass to draw a circle. Label the center. Use a straightedge to draw a central angle. Label it. raw points Locate three points on ( in the eterior of and label them T, U, and V. Measure angles raw T, U, and V. These are called inscribed angles. Measure each angle. at classzone.com W ONLUION Use your observations to complete these eercises 1. opy and complete the table. entral angle Inscribed angle 1 Inscribed angle 2 Inscribed angle 3 Name T U V Measure???? 2. raw two more circles. epeat teps 1 3 using different central angles. ecord the measures in a table similar to the one above. 3. Use your results to make a conjecture about how the measure of an inscribed angle is related to the measure of the corresponding central angle Use Inscribed ngles and olygons 671

25 10.4 Use Inscribed ngles and olygons efore You used central angles of circles. Now You will use inscribed angles of circles. Why? o you can take a picture from multiple angles, as in ample 4. Key Vocabulary inscribed angle intercepted arc inscribed polygon circumscribed circle n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. inscribed angle intercepted arc THOM or Your Notebook THOM 10.7 Measure of an Inscribed ngle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc. roof: s , p. 678 m 5 1 } 2 m The proof of Theorem 10.7 in ercises involves three cases. ase 1 enter is on a side of the inscribed angle. ase 2 enter is inside the inscribed angle. ase 3 enter is outside the inscribed angle. XML 1 Use inscribed angles ind the indicated measure in (. a. m T b. m Q olution a. m T 5 } 1 2 m 5 1 }2 (488) T b. m TQ 5 2m 5 2 p ecause TQ is a semicircle, m Q m TQ o, m Q hapter 10 roperties of ircles

26 XML 2 ind the measure of an intercepted arc ind m and m T. What do you notice about T and U? T 318 U olution rom Theorem 10.7, you know that m 5 2m U 5 2(318) lso, m T 5 } 1 2 m 5 1 }2 (628) o, T > U. INTTING TH M ample 2 suggests Theorem THOM or Your Notebook THOM 10.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. roof:. 34, p. 678 > XML 3 tandardized Test ractice Name two pairs of congruent angles in the figure. JKM > KJL, JLM > KML JLM > KJL, JKM > KML J K JKM > JLM, KJL > KML JLM > KJL, JLM > JKM M L LIMINT HOI You can eliminate choices and, because they do not include the pair JKM > JLM. olution Notice that JKM and JLM intercept the same arc, and so JKM > JLM by Theorem lso, KJL and KML intercept the same arc, so they must also be congruent. Only choice contains both pairs of angles. c o, by Theorem 10.8, the correct answer is. GUI TI for amples 1, 2, and 3 ind the measure of the red arc or angle. 1. G H T V 388 U 3. Z Y 728 X W 10.4 Use Inscribed ngles and olygons 673

27 OLYGON polygon is an inscribed polygon if all of its vertices lie on a circle. The circle that contains the vertices is a circumscribed circle. inscribed triangle circumscribed circles inscribed quadrilateral THOM THOM 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. roof:. 35, p. 678 or Your Notebook m if and only if } is a diameter of the circle. XML 4 Use a circumscribed circle HOTOGHY Your camera has a 908 field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way? olution rom Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. o, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera s 908 field of vision from any point on the semicircle in front of the statue. GUI TI for ample 4 4. WHT I? In ample 4, eplain how to find locations if you want to frame the front and left side of the statue in your picture. 674 hapter 10 roperties of ircles

28 INI QUILTL Only certain quadrilaterals can be inscribed in a circle. Theorem describes these quadrilaterals. THOM or Your Notebook THOM quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.,,, and G lie on ( if and only if m 1 m 5 m 1 m G roof:. 30, p. 678; p. 938 G XML 5 Use Theorem ind the value of each variable. a y8 8 b. K L 2a8 4b8 2a8 J 2b8 M olution a. Q is inscribed in a circle, so opposite angles are supplementary. m 1 m m Q 1 m y y b. JKLM is inscribed in a circle, so opposite angles are supplementary. m J 1 m L m K 1 m M a8 1 2a b812b a b a 5 45 b 5 30 GUI TI for ample 5 ind the value of each variable y c8 T U V (2c 2 6) Use Inscribed ngles and olygons 675

29 10.4 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 11, 13, and 29 5 TNIZ TT TI s. 2, 16, 18, 29, and VOULY opy and complete: If a circle is circumscribed about a polygon, then the polygon is? in the circle. 2. WITING plain why the diagonals of a rectangle inscribed in a circle are diameters of the circle. XML 1 and 2 on pp for s. 3 9 INI NGL ind the indicated measure. 3. m 4. m G 5. m N G 1208 N 1608 L M 6. m 7. m VU 8. mwx Y 678 T 308 U W V X 9. O NLYI escribe the error in the diagram of (. ind two ways to correct the error. Q 45º 100º XML 3 ONGUNT NGL Name two pairs of congruent angles. on p. 673 for s K 12. W J M L X Y Z XML 5 LG ind the values of the variables. on p. 675 for s y m k8 608 G 15. J 548 M 4b8 K 3a T 1308 L 676 hapter 10 roperties of ircles

30 16. MULTIL HOI In the diagram, is a central angle and m What is m? INI NGL In each star below, all of the inscribed angles are congruent. ind the measure of an inscribed angle for each star. Then find the sum of all the inscribed angles for each star. a. b. c. 18. MULTIL HOI What is the value of? ( )8 (8 1 10)8 G 19. LLLOGM arallelogram QT is inscribed in (. ind m. ONING etermine whether the quadrilateral can always be inscribed in a circle. plain your reasoning. 20. quare 21. ectangle 22. arallelogram 23. Kite 24. hombus 25. Isosceles trapezoid 26. HLLNG In the diagram, is a right angle. If you draw the smallest possible circle through and tangent to }, the circle will intersect } at J and } at K. ind the eact length of } JK OLM OLVING 27. TONOMY uppose three moons,, and orbit 100,000 kilometers above the surface of a planet. uppose m 5 908, and the planet is 20,000 kilometers in diameter. raw a diagram of the situation. How far is moon from moon? XML 4 on p. 674 for NT carpenter s square is an L-shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use a carpenter s square to draw a diameter on the circular piece of wood? 10.4 Use Inscribed ngles and olygons 677

31 29. WITING right triangle is inscribed in a circle and the radius of the circle is given. plain how to find the length of the hypotenuse. 30. OVING THOM opy and complete the proof that opposite angles of an inscribed quadrilateral are supplementary. GIVN c ( with inscribed quadrilateral G OV c m 1 m , m 1 m G y the rc ddition ostulate, m G 1? and m G 1 m Using the? Theorem, m G 5 2m, m G 5 2m, m 5 2m G, and m G 5 2m. y the ubstitution roperty, 2m 1? , so?. imilarly,?. G OVING THOM 10.7 If an angle is inscribed in (Q, the center Q can be on a side of the angle, in the interior of the angle, or in the eterior of the angle. In ercises 31 33, you will prove Theorem 10.7 for each of these cases. 31. ase 1 rove ase 1 of Theorem GIVN c is inscribed in (Q. Let m 5 8. oint Q lies on }. OV c m 5 1 } 2 m lan for roof how that n Q is isosceles. Use the ase ngles Theorem and the terior ngles Theorem to show that m Q Then, show that m olve for, and show that m 5 1 } 2 m ase 2 Use the diagram and auiliary line to write GIVN and OV statements for ase 2 of Theorem Then write a plan for proof. 33. ase 3 Use the diagram and auiliary line to write GIVN and OV statements for ase 3 of Theorem Then write a plan for proof. 34. OVING THOM 10.8 Write a paragraph proof of Theorem irst draw a diagram and write GIVN and OV statements. 35. OVING THOM 10.9 Theorem 10.9 is written as a conditional statement and its converse. Write a plan for proof of each statement. 36. XTN ON In the diagram, ( and (M intersect at, and } is a diameter of (M. plain why ] is tangent to (. M WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

32 HLLNG In ercises 37 and 38, use the following information. You are making a circular cutting board. To begin, you glue eight 1 inch by 2 inch boards together, as shown at the right. Then you draw and cut a circle with an 8 inch diameter from the boards. 37. } H is a diameter of the circular cutting board. Write a proportion relating GJ and JH. tate a theorem to justify your answer. 38. ind J, JH, and JG. What is the length of the cutting board seam labeled } GK? L M J G K H 39. HUTTL To maimize thrust on a N space shuttle, engineers drill an 11-point star out of the solid fuel that fills each booster. They begin by drilling a hole with radius 2 feet, and they would like each side of the star to be 1.5 feet. Is this possible if the fuel cannot have angles greater than 458 at its points? 1.5 ft 2 ft MIX VIW VIW repare for Lesson 10.5 in s ind the approimate length of the hypotenuse. ound your answer to the nearest tenth. (p. 433) Graph the reflection of the polygon in the given line. (p. 589) 43. y-ais y 5 2 y y y G ΠH 1 1 ketch the image of (3, 24) after the described glide reflection. (p. 608) 46. Translation: (, y) (, y 2 2) 47. Translation: (, y) ( 1 1, y 1 4) eflection: in the y-ais eflection: in y 5 4 XT TI for Lesson 10.4, p. 915 ONLIN QUIZ at classzone.com 679

33 10.5 pply Other ngle elationships in ircles efore You found the measures of angles formed on a circle. Now You will find the measures of angles inside or outside a circle. Why o you can determine the part of arth seen from a hot air balloon, as in. 25. Key Vocabulary chord, p. 651 secant, p. 651 tangent, p. 651 You know that the measure of an inscribed angle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. THOM or Your Notebook THOM If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. roof:. 27, p. 685 m } 2 m m }2 m 2 1 XML 1 ind angle and arc measures Line m is tangent to the circle. ind the measure of the red angle or arc. a. b. K m m J 1258 L olution a. m } 2 (1308) b. mkjl 5 2(1258) GUI TI for ample 1 ind the indicated measure. 1. m 1 2. m T 3. m XY T 988 Y 808 X 680 hapter 10 roperties of ircles

34 INTTING LIN N IL If two lines intersect a circle, there are three places where the lines can intersect. on the circle inside the circle outside the circle You can use Theorems and to find measures when the lines intersect inside or outside the circle. THOM THOM ngles Inside the ircle Theorem If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. roof:. 28, p. 685 or Your Notebook 1 2 m 1 5 } 1 1 m 2 1 m 2, m 2 5 } 1 1 m 2 1 m 2 THOM ngles Outside the ircle Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 2 m } 2 1 m 2 m 2 m } 2 1m Q 2 m 2 m }2 1 m XY 2 m WZ 2 roof:. 29, p. 685 W 3 Z X Y XML 2 ind an angle measure inside a circle ind the value of M olution J 8 L The chords } JL and } KM intersect inside the circle. K } 2 1mJM 1 mlk 2 Use Theorem } 2 ( ) ubstitute. implify pply Other ngle elationships in ircles 681

35 XML 3 ind an angle measure outside a circle ind the value of. 8 olution The tangent ] and the secant ] intersect outside the circle. m 5 } 1 1 m 2 m 2 Use Theorem } 2 ( ) 5 51 ubstitute. implify. XML 4 olve a real-world problem IN The Northern Lights are bright flashes of colored light between 50 and 200 miles above arth. uppose a flash occurs 150 miles above arth. What is the measure of arc, the portion of arth from which the flash is visible? (arth s radius is approimately 4000 miles.) 4150 mi 4000 mi olution ecause } and } are tangents, } } and } }. lso, } > } and } > }.o,n > n by the Hypotenuse-Leg ongruence Theorem, and >. olve right n to find that m ø o, m ø 2(74.58) ø Let m 5 8. m 5 1 } 2 1 m 2 m 2 Use Theorem Not drawn to scale VOI O ecause the value for m is an approimation, use the symbol ø instead of ø 1 } 2 [(360828) 2 8] ubstitute. ø 31 olve for. c The measure of the arc from which the flash is visible is about 318. at classzone.com GUI TI for amples 2, 3, and 4 ind the value of the variable. 4. y J K 448 H 308 G a T 3 U hapter 10 roperties of ircles

36 10.5 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 3, 9, and 23 5 TNIZ TT TI s. 2, 6, 13, 15, 19, and VOULY opy and complete: The points,,, and are on a circle and ] intersects ] at. If m 5 1 } 2 1m 2 m 2, then is? (inside, on, or outside) the circle. 2. WITING What does it mean in Theorem if m 5 08? Is this consistent with what you learned in Lesson 10.4? plain your answer. XML 1 on p. 680 for s. 3 6 INING MU Line t is tangent to the circle. ind the indicated measure. 3. m 4. m 5. m 1 t t 658 t MULTIL HOI The diagram at the right is not drawn to scale. } is any chord that is not a diameter of the circle. Line m is tangent to the circle at point. Which statement must be true? m 5 90 Þ 90 INING MU ind the value of. XML 2 on p. 681 for s H G 9. J 308 M 8 K L (2 2 30)8 XML 3 on p. 682 for s G U 348 T ( 1 6)8 (3 2 2)8 W V 13. MULTIL HOI In the diagram, l is tangent to the circle at. Which relationship is not true? m m m m T l 10.5 pply Other ngle elationships in ircles 683

37 14. O NLYI escribe the error in the diagram below HOT ON In the diagram at the right, ] L is tangent to the circle and } KJ is a diameter. What is the range of possible angle measures of LJ? plain. K L 16. ONNTI IL The circles below are concentric. a. ind the value of. b. press c in terms of a and b. J a8 b8 c8 17. INI IL In the diagram, the circle is inscribed in nq. ind m, m G,andm G. 18. LG In the diagram, ] is tangent to (. ind m. 408 G WITING oints and are on a circle and t is a tangent line containing and another point. a. raw two different diagrams that illustrate this situation. b. Write an equation for m in terms of m for each diagram. c. When will these equations give the same value for m? HLLNG ind the indicated measure(s). 20. ind m if mwzy ind m and m. Z W Y X J G 1158 H WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

38 OLM OLVING VIO OING In the diagram at the right, television cameras are positioned at,, and to record what happens on stage. The stage is an arc of (. Use the diagram for ercises ind m, m, and m. 23. The wall is tangent to the circle. ind without using the measure of. 24. You would like amera to have a 308 view of the stage. hould you move the camera closer or further away from the stage? plain. XML 4 on p. 682 for HOT I LLOON You are flying in a hot air balloon about 1.2 miles above the ground. Use the method from ample 4 to find the measure of the arc that represents the part of arth that you can see. The radius of arth is about 4000 miles. 26. XTN ON cart is resting on its handle. The angle between the handle and the ground is 148 and the handle connects to the center of the wheel. What are the measures of the arcs of the wheel between the ground and the cart? plain. 27. OVING THOM The proof of Theorem can be split into three cases. The diagram at the right shows the case where } contains the center of the circle. Use Theorem 10.1 to write a paragraph proof for this case. What are the other two cases? (Hint: ee ercises on page 678.) raw a diagram and write plans for proof for the other cases. 28. OVING THOM Write a proof of Theorem GIVN c hords } and } intersect. OV c m } 2 1 m 1 m OVING THOM Use the diagram at the right to prove Theorem for the case of a tangent and a secant. raw }. plain how to use the terior ngle Theorem in the proof of this case. Then copy the diagrams for the other two cases from page 681, draw appropriate auiliary segments, and write plans for proof for these cases pply Other ngle elationships in ircles 685

39 30. OO Q and are points on a circle. is a point outside the circle. } Q and } are tangents to the circle. rove that } Q is not a diameter. 31. HLLNG block and tackle system composed of two pulleys and a rope is shown at the right. The distance between the centers of the pulleys is 113 centimeters and the pulleys each have a radius of 15 centimeters. What percent of the circumference of the bottom pulley is not touching the rope? MIX VIW lassify the dilation and find its scale factor. (p. 626) VIW repare for Lesson 10.6 in s Use the quadratic formula to solve the equation. ound decimal answers to the nearest hundredth. (pp. 641, 883) QUIZ for Lessons ind the value(s) of the variable(s). 1. m 5 z8 (p. 672) 2. m GH 5 z8 (p. 672) 3. m JKL 5 z8 (p. 672) K G 8 y8 J (11 1 y)8 H M L (p. 680) (p. 680) (p. 680) 7. MOUNTIN You are on top of a mountain about 1.37 miles above sea level. ind the measure of the arc that represents the part of arth that you can see. arth s radius is approimately 4000 miles. (p. 680) 686 XT TI for Lesson 10.5, p. 915 ONLIN QUIZ at classzone.com

40 MIX VIW of roblem olving Lessons MULTI-T OLM n official stands 2 meters from the edge of a discus circle and 3 meters from a point of tangency. TT TT TI classzone.com 4. XTN ON The Navy ier erris Wheel in hicago is 150 feet tall and has 40 spokes. 2 m 3 m a. ind the radius of the discus circle. b. How far is the official from the center of the discus circle? 2. GI NW In the diagram, } XY > } YZ and m XQZ ind m YZ in degrees. Y X Z 3. MULTI-T OLM wind turbine has three equally spaced blades that are each 131 feet long. a. ind the measure of the angle between any two spokes. b. Two spokes form a central angle of 728. How many spokes are between the two spokes? c. The bottom of the wheel is 10 feet from the ground. ind the diameter and radius of the wheel. plain your reasoning. 5. ON-N raw a quadrilateral inscribed in a circle. Measure two consecutive angles. Then find the measures of the other two angles algebraically. 6. MULTI-T OLM Use the diagram. y 938 L 8 M 358 N a. What is the measure of the arc between any two blades? b. The highest point reached by a blade is 361 feet above the ground. ind the distance between the lowest point reached by the blades and the ground. c. What is the distance y from the tip of one blade to the tip of another blade? ound your answer to the nearest tenth. K a. ind the value of. b. ind the measures of the other three angles formed by the intersecting chords. 7. HOT ON Use the diagram to show that m 5 y y8 Mied eview of roblem olving 687

41 TIVITY Investigating Geometry Use before Lesson Investigate egment Lengths MTIL graphing calculator or computer classzone.com Keystrokes QUTION What is the relationship between the lengths of segments in a circle? You can use geometry drawing software to find a relationship between the segments formed by two intersecting chords. XLO raw a circle with two chords =6.93 = T 1 raw a circle raw a circle and choose four points on the circle. Label them,,, and. T 2 raw secants raw secants ] and ] and label the intersection point. T 3 Measure segments Note that } and } are chords. Measure }, }, }, and } in your diagram. T 4 erform calculations alculate the products p and p. W ONLUION Use your observations to complete these eercises 1. What do you notice about the products you found in tep 4? 2. rag points,,, and, keeping point inside the circle. What do you notice about the new products from tep 4? 3. Make a conjecture about the relationship between the four chord segments. 4. Let } Q and } be two chords of a circle that intersect at the point T. If T 5 9, QT 5 5, and T 5 15, use your conjecture from ercise 3 to find T. 688 hapter 10 roperties of ircles

42 10.6 ind egment Lengths in ircles efore You found angle and arc measures in circles. Now You will find segment lengths in circles. Why? o you can find distances in astronomy, as in ample 4. Key Vocabulary segments of a chord secant segment eternal segment When two chords intersect in the interior of a circle, each chord is divided into two segments that are called segments of the chord. THOM or Your Notebook THOM egments of hords Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. roof:. 21, p. 694 p 5 p lan for roof To prove Theorem 10.14, construct two similar triangles. The lengths of the corresponding sides are proportional, so } 5 }. y the ross roducts roperty, p 5 p. XML 1 ind lengths using Theorem LG ind ML and JK. M olution NK p NJ 5 NL p NM Use Theorem p ( 1 4) 5 ( 1 1) p ( 1 2) ubstitute implify. K 1 2 N L J ubtract 2 from each side. 5 2 olve for. ind ML and JK by substitution. ML 5 ( 1 2) 1 ( 1 1) JK 5 1 ( 1 4) ind egment Lengths in ircles 689

43 TNGNT N NT secant segment is a segment that contains a chord of a circle, and has eactly one endpoint outside the circle. The part of a secant segment that is outside the circle is called an eternal segment. eternal segment secant segment tangent segment THOM or Your Notebook THOM egments of ecants Theorem If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its eternal segment equals the product of the lengths of the other secant segment and its eternal segment. roof:. 25, p. 694 p 5 p XML 2 tandardized Test ractice What is the value of? } T olution Q p 5 p T Use Theorem p (5 1 4) 5 3 p ( 1 3) ubstitute implify. 95 olve for. c The correct answer is. GUI TI for amples 1 and 2 ind the value(s) of hapter 10 roperties of ircles

44 THOM or Your Notebook THOM egments of ecants and Tangents Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its eternal segment equals the square of the length of the tangent segment. roof:. 26, p p XML 3 ind lengths using Theorem NOTH WY or an alternative method for solving the problem in ample 3, turn to page 696 for the roblem olving Workshop. Use the figure at the right to find. olution Q 2 5 p T Use Theorem p ( 1 8) ubstitute implify Write in standard form T 28 6 Ï}} (1)(2256) Use quadratic formula. 2(1) Ï } 17 implify. Use the positive solution, because lengths cannot be negative. c o, Ï } 17 < 12.49, and < at classzone.com GUI TI for ample 3 ind the value of etermine which theorem you would use to find. Then find the value of In the diagram for Theorem 10.16, what must be true about compared to? 10.6 ind egment Lengths in ircles 691

45 XML 4 olve a real-world problem IN Tethys, alypso, and Telesto are three of aturn s moons. ach has a nearly circular orbit 295,000 kilometers in radius. The assini-huygens spacecraft entered aturn s orbit in July Telesto is on a point of tangency. ind the distance from assini to Tethys. olution p 5 2 Use Theorem ,000 p < 203,000 2 ubstitute. < 496,494 olve for. c assini is about 496,494 kilometers from Tethys. GUI TI for ample Why is it appropriate to use the approimation symbol < in the last two steps of the solution to ample 4? 10.6 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 3, 9, and 21 5 TNIZ TT TI s. 2, 16, 24, and VOULY opy and complete: The part of the secant segment that is outsidethecircleiscalleda(n)?. 2. WITING plain the difference between a tangent segment and a secant segment. XML 1 INING GMNT LNGTH ind the value of. on p. 689 for s hapter 10 roperties of ircles

46 INING GMNT LNGTH ind the value of. XML 2 on p. 690 for s XML 3 on p. 691 for s O NLYI escribe and correct the error in finding. p 5 p p p 3 p INING GMNT LNGTH ind the value of. ound to the nearest tenth MULTIL HOI Which of the following is a possible value of? INING LNGTH ind Q. ound your answers to the nearest tenth. 17. N 6 12 M HLLNG In the figure, 5 12, 5 8, 5 6, 5 4, and is a point of tangency. ind the radius of ( ind egment Lengths in ircles 693

47 OLM OLVING XML 4 on p. 692 for HOLOGY The circular stone mound in Ireland called Newgrange has a diameter of 250 feet. passage 62 feet long leads toward the center of the mound. ind the perpendicular distance from the end of the passage to either side of the mound. 21. OVING THOM Write a two-column proof of Theorem Use similar triangles as outlined in the lan for roof on page WLL In the diagram of the water well,,, and are known. Write an equation for using these three measurements. 23. OO Use Theorem 10.1 to prove Theorem for the special case when the secant segment contains the center of the circle. G 24. HOT ON You are designing an animated logo for your website. parkles leave point and move to the circle along the segments shown so that all of the sparkles reach the circle at the same time. parkles travel from point to point at 2 centimeters per second. How fast should sparkles move from point to point N? plain. 25. OVING THOM Use the plan to prove Theorem GIVN c } and } are secant segments. OV c p 5 p lan for roof raw } and }. how that n and n are similar. Use the fact that corresponding side lengths in similar triangles are proportional. 26. OVING THOM Use the plan to prove Theorem GIVN c } is a tangent segment. } is a secant segment. OV c 2 5 p lan for roof raw } and }. Use the fact that corresponding side lengths in similar triangles are proportional WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

48 27. XTN ON In the diagram, } is a tangent segment, m , m 5 208, m 5 608, 5 6, 5 3, and a. ind m. b. how that n, n. c. Let 5 y and 5. Use the results of part (b) to write a proportion involving and y. olve for y. d. Use a theorem from this section to write another equation involving both and y. e. Use the results of parts (c) and (d) to solve for and y. f. plain how to find HLLNG tereographic projection is a map-making technique that takes points on a sphere with radius one unit (arth) to points on a plane (the map). The plane is tangent to the sphere at the origin. quator N y The map location for each point on the sphere is found by etending the line that connects N and. The point s projection is where the line intersects the plane. ind the distance d from the point to its corresponding point 9(4, 23) on the plane. quator Not drawn to scale (0, 0) d (4, 3) MIX VIW VIW repare for Lesson 10.7 in s valuate the epression. (p. 874) 29. Ï }} (210) Ï }} 25 1 (24) 1 (6 2 1) Ï }}} [22 2 (26)] 2 1 (3 2 6) In right n Q, Q 5 8, m Q 5 408, and m ind Q and to the nearest tenth. (p. 473) 33. ] is tangent to ( at. The radius of ( is 5 and 5 8. ind. (p. 651) ind the indicated measure. } and } are diameters. (p. 659) 34. m 35. m 36. m 37. m 38. m 39. m etermine whether } is a diameter of the circle. plain. (p. 664) XT TI for Lesson 10.6, p. 915 ONLIN QUIZ at classzone.com 695

49 LON 10.6 Using LTNTIV MTHO nother Way to olve ample 3, page 691 MULTIL NTTION You can use similar triangles to find the length of an eternal secant segment. OLM Use the figure at the right to find T M THO Using imilar Triangles T 1 raw segments } Q and } QT, and identify the similar triangles. ecause they both intercept the same arc, Q > TQ. y the efleive roperty of ngle ongruence, Q > TQ. o, nq, nqt by the imilarity ostulate. T 2 Use a proportion to solve for. } Q 5 Q } T } } 1 8 c y the ross roducts roperty, Use the quadratic formula to find that Ï } 17. Taking the positive solution, Ï } 17 and TI 1. WHT I? ind Q in the problem above if the known lengths are 5 4 and T MULTI-T OLM opy the diagram. 3. HO ind the value of GMNT O NT Use the egments of ecants Theorem to write an epression for w in terms of, y, and z. a. raw auiliary segments } and }. Name two similar triangles. b. If 5 15, 5 5, and 5 12, find. w z y 696 hapter 10 roperties of ircles

50 tension Use after Lesson 10.6 raw a Locus GOL raw the locus of points satisfying certain conditions. Key Vocabulary locus locus in a plane is the set of all points in a plane that satisfy a given condition or a set of given conditions. The word locus is derived from the Latin word for location. The plural of locus is loci, pronounced low-sigh. locus is often described as the path of an object moving in a plane. or eample, the reason that many clock faces are circular is that the locus of the end of a clock s minute hand is a circle. XML 1 ind a locus raw a point on a piece of paper. raw and describe the locus of all points on the paper that are 1 centimeter from. olution T 1 T 2 T 3 raw point. Locate several points 1 centimeter from. ecognize a pattern: the points lie on a circle. raw the circle. c The locus of points on the paper that are 1 centimeter from is a circle with center and radius 1 centimeter. KY ONT or Your Notebook How to ind a Locus To find the locus of points that satisfy a given condition, use the following steps. T 1 raw any figures that are given in the statement of the problem. Locate several points that satisfy the given condition. T 2 ontinue drawing points until you can recognize the pattern. T 3 raw the locus and describe it in words. tension: Locus 697

51 LOI TIYING TWO O MO ONITION To find the locus of points that satisfy two or more conditions, first find the locus of points that satisfy each condition alone. Then find the intersection of these loci. XML 2 raw a locus satisfying two conditions oints and lie in a plane. What is the locus of points in the plane that are equidistant from points and and are a distance of from? olution T 1 T 2 T 3 The locus of all points that are equidistant from and is the perpendicular bisector of }. The locus of all points that are a distance of from is the circle with center and radius. These loci intersect at and. o and form the locus of points that satisfy both conditions. TI XML 1 on p. 697 for s. 1 4 XML 2 on p. 698 for s. 5 9 WING LOU raw the figure. Then sketch the locus of points on the paper that satisfy the given condition. 1. oint, the locus of points that are 1 inch from 2. Line k, the locus of points that are 1 inch from k 3. oint, the locus of points that are at least 1 inch from 4. Line j, the locus of points that are no more than 1 inch from j WITING Write a description of the locus. Include a sketch. 5. oint lies on line l. What is the locus of points on l and 3 cm from? 6. oint Q lies on line m. What is the locus of points 5 cm from Q and 3 cm from m? 7. oint is 10 cm from line k. What is the locus of points that are within 10 cm of, but further than 10 cm from k? 8. Lines l and m are parallel. oint is 5 cm from both lines. What is the locus of points between l and m and no more than 8 cm from? 9. OG LH dog s leash is tied to a stake at the corner of its doghouse, as shown at the right. The leash is 9 feet long. Make a scale drawing of the doghouse and sketch the locus of points that the dog can reach. 698 hapter 10 roperties of ircles

52 10.7 Write and Graph quations of ircles efore You wrote equations of lines in the coordinate plane. Now You will write equations of circles in the coordinate plane. Why? o you can determine zones of a commuter system, as in. 36. Key Vocabulary standard equation of a circle Let (, y) represent any point on a circle with center at the origin and radius r. y the ythagorean Theorem, 2 1 y 2 5 r 2. This is the equation of a circle with radius r and center at the origin. r y (, y) y XML 1 Write an equation of a circle Write the equation of the circle shown. y olution The radius is 3 and the center is at the origin. 2 1 y 2 5 r 2 quation of circle y ubstitute. 2 1 y implify. c The equation of the circle is 2 1 y IL NT T (h, k) You can write the equation of any circle if you know its radius and the coordinates of its center. y uppose a circle has radius r and center (h, k). Let (, y) (, y) be a point on the circle. The distance between (, y) and r (h, k) is r, so by the istance ormula (h, k) Ï }} ( 2 h) 2 1 (y 2 k) 2 5 r. quare both sides to find the standard equation of a circle. KY ONT or Your Notebook tandard quation of a ircle The standard equation of a circle with center (h, k) and radius r is: ( 2 h) 2 1 (y 2 k) 2 5 r Write and Graph quations of ircles 699

53 XML 2 Write the standard equation of a circle Write the standard equation of a circle with center (0, 29) and radius 4.2. olution ( 2 h) 2 1 (y 2 k) 2 5 r 2 tandard equation of a circle ( 2 0) 2 1 (y 2 (29)) ubstitute. 2 1 (y 1 9) implify. GUI TI for amples 1 and 2 Write the standard equation of the circle with the given center and radius. 1. enter (0, 0), radius enter (22, 5), radius 7 XML 3 Write the standard equation of a circle The point (25, 6) is on a circle with center (21, 3). Write the standard equation of the circle. (25, 6) y olution To write the standard equation, you need to know the values of h, k, and r. To find r, find the distance between the center and the point (25, 6) on the circle. r 5 Ï }}} [25 2 (21)] 2 1 (6 2 3) 2 istance ormula (21, 3) Ï } (24) implify. 5 5 implify. ubstitute (h, k) 5 (21, 3) and r 5 5 into the standard equation of a circle. ( 2 h) 2 1 (y 2 k) 2 5 r 2 tandard equation of a circle [ 2 (21)] 2 1 (y 2 3) ubstitute. ( 1 1) 2 1 (y 2 3) implify. c The standard equation of the circle is ( 1 1) 2 1 (y 2 3) GUI TI for ample 3 3. The point (3, 4) is on a circle whose center is (1, 4). Write the standard equation of the circle. 4. The point (21, 2) is on a circle whose center is (2, 6). Write the standard equation of the circle. 700 hapter 10 roperties of ircles

54 XML 4 Graph a circle U QUTION If you know the equation of a circle, you can graph the circle by identifying its center and radius. The equation of a circle is ( 2 4) 2 1 (y 1 2) Graph the circle. olution ewrite the equation to find the center and radius. ( 2 4) 2 1 (y 1 2) ( 2 4) 2 1 [y 2 (22)] The center is (4, 22) and the radius is 6. Use a compass to graph the circle. 4 y 2 (4, 22) XML 5 Use graphs of circles THQUK The epicenter of an earthquake is the point on arth s surface directly above the earthquake s origin. seismograph can be used to determine the distance to the epicenter of an earthquake. eismographs are needed in three different places to locate an earthquake s epicenter. Use the seismograph readings from locations,, and to find the epicenter of an earthquake. The epicenter is 7 miles away from (22, 2.5). The epicenter is 4 miles away from (4, 6). The epicenter is 5 miles away from (3, 22.5). olution The set of all points equidistant from a given point is a circle, so the epicenter is located on each of the following circles. ( with center (22, 2.5) and radius 7 ( with center (4, 6) and radius y ( with center (3, 22.5) and radius 5 To find the epicenter, graph the circles on a graph where units are measured in miles. ind the point of intersection of all three circles c The epicenter is at about (5, 2). at classzone.com GUI TI for amples 4 and 5 5. The equation of a circle is ( 2 4) 2 1 (y 1 3) Graph the circle. 6. The equation of a circle is ( 1 8) 2 1 (y 1 5) Graph the circle. 7. Why are three seismographs needed to locate an earthquake s epicenter? 10.7 Write and Graph quations of ircles 701

55 10.7 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 7, 17, and 37 5 TNIZ TT TI s. 2, 16, 26, and VOULY opy and complete: The standard equation of a circle can be written for any circle with known? and?. 2. WITING plain why the location of the center and one point on a circle is enough information to draw the rest of the circle. XML 1 and 2 on pp for s WITING QUTION Write the standard equation of the circle. 3. y 4. y 5. y y 7. y 8. 3 y WITING QUTION Write the standard equation of the circle with the given center and radius. 9. enter (0, 0), radius enter (24, 1), radius enter (7, 26), radius enter (4, 1), radius enter (3, 25), radius enter (23, 4), radius O NLYI escribe and correct the error in writing the equation of a circle. n equation of a circle with center (23, 25) and radius 3 is ( 2 3) 2 1 (y 2 5) MULTIL HOI The standard equation of a circle is ( 2 2) 2 1 (y 1 1) What is the diameter of the circle? XML 3 on p. 700 for s WITING QUTION Use the given information to write the standard equation of the circle. 17. The center is (0, 0), and a point on the circle is (0, 6). 18. The center is (1, 2), and a point on the circle is (4, 2). 19. The center is (23, 5), and a point on the circle is (1, 8). 702 hapter 10 roperties of ircles

56 XML 4 on p. 701 for s GHING IL Graph the equation y ( 2 3) 2 1 y (y 1 2) ( 2 4) 2 1 (y 2 1) ( 1 5) 2 1 (y 2 3) ( 1 2) 2 1 (y 1 6) MULTIL HOI Which of the points does not lie on the circle described by the equation ( 1 2) 2 1 (y 2 4) ? (22, 21) (1, 8) (3, 4) (0, 5) LG etermine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form y 2 2 6y y 2 1 2y y 2 1 4y y INTIYING TY O LIN Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. 31. ircle: ( 2 4) 2 1 (y 2 3) ircle: ( 1 2) 2 1 (y 2 2) Line: y Line: y ircle: ( 2 5) 2 1 (y 1 1) ircle: ( 1 3) 2 1 (y 2 6) Line: y 5 1 } Line: y 52 4 } HLLNG our tangent circles are centered on the -ais. The radius of ( is twice the radius of (O. The radius of ( is three times the radius of (O. The radius of ( is four times the radius of (O. ll circles have integer radii and the point (63, 16) is on (. What is the equation of (? y O OLM OLVING XML 5 on p. 701 for OMMUT TIN city s commuter system has three zones covering the regions described. Zone 1 covers people living within three miles of the city center. Zone 2 covers those between three and seven miles from the center, and Zone 3 covers those over seven miles from the center. a. Graph this situation with the city center at the origin, where units are measured in miles. b. ind which zone covers people living at (3, 4), (6, 5), (1, 2), (0, 3), and (1, 6). 40 Zone 3 Zone 1 87 Zone mi 10.7 Write and Graph quations of ircles 703

57 37. OMT I The diameter of a is about 4.8 inches. The diameter of the hole in the center is about 0.6 inches. You place a on the coordinate plane with center at (0, 0). Write the equations for the outside edge of the disc and the edge of the hole in the center. 4.8 in. 0.6 in. ULUX OLYGON In ercises 38 41, use the following information. The figure at the right is called a euleau polygon. It is not a true polygon because its sides are not straight. n is equilateral. 38. J lies on a circle with center and radius. Write an equation of this circle. 39. lies on a circle with center and radius. Write an equation of this circle. y 40. ONTUTION The remaining arcs of the polygon are constructed in the same way as J and in ercises 38 and 39. onstruct a euleau polygon on a piece of cardboard. 41. ut out the euleau polygon from ercise 40. oll it on its edge like a wheel and measure its height when it is in different orientations. plain why a euleau polygon is said to have constant width. 42. XTN ON Telecommunication towers can be used to transmit cellular phone calls. Towers have a range of about 3 km. graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). a. raw the graph and locate the towers. re there any areas that may receive calls from more than one tower? b. uppose your home is located at (2, 6) and your school is at (2.5, 3). an you use your cell phone at either or both of these locations? c. ity is located at (22, 2.5) and ity is at (5, 4). ach city has a radius of 1.5 km. Which city seems to have better cell phone coverage? plain. J 1 1 H G 43. ONING The lines y 5 } and y 52} are tangent to ( at the points (4, 5) and (4, 13), respectively. a. ind the coordinates of and the radius of (. plain your steps. b. Write the standard equation of ( and draw its graph. 44. OO Write a proof. y GIVN c circle passing through the points (21, 0) and (1, 0) OV c The equation of the circle is 2 2 2yk 1 y with center at (0, k). (21, 0) (1, 0) WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

58 45. HLLNG The intersecting lines m and n are tangent to ( at the points (8, 6) and (10, 8), respectively. a. What is the intersection point of m and n if the radius r of ( is 2? What is their intersection point if r is 10? What do you notice about the two intersection points and the center? b. Write the equation that describes the locus of intersection points of m and n for all possible values of r. MIX VIW VIW repare for Lesson 11.1 in s ind the perimeter of the figure. 46. (p. 49) 47. (p. 49) 48. (p. 433) 9 in. 18 ft 40 m 22 in. 57 m ind the circumference of the circle with given radius r or diameter d. Use p (p. 49) 49. r 5 7 cm 50. d in. 51. d 5 48 yd ind the radius r of (. (p. 651) r r 9 r r r r QUIZ for Lessons ind the value of. (p. 689) In ercises 4 and 5, use the given information to write the standard equation of the circle. (p. 699) 4. The center is (1, 4), and the radius is The center is (5, 27), and a point on the circle is (5, 23). 6. TI The diameter of a certain tire is 24.2 inches. The diameter of the rim in the center is 14 inches. raw the tire in a coordinate plane with center at (24, 3). Write the equations for the outer edge of the tire and for the rim where units are measured in inches. (p. 699) XT TI for Lesson 10.7, p. 915 ONLIN QUIZ at classzone.com 705

59 MIX VIW of roblem olving Lessons HOT ON local radio station can broadcast its signal 20 miles. The station is located at the point (20, 30) where units are measured in miles. a. Write an inequality that represents the area covered by the radio station. b. etermine whether you can receive the radio station s signal when you are located at each of the following points: (25, 25), (10, 10), G(20, 16), and H(35, 30). 4. HOT ON You are at point, about 6 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is 17 feet. 17 ft 6 ft r ft r ft TT TT TI classzone.com 2. XTN ON ell phone towers are used to transmit calls. n area has cell phone towers at points (2, 3), (4, 5), and (5, 3) where units are measured in miles. ach tower has a transmission radius of 2 miles. a. raw the area on a graph and locate the three cell phone towers. re there any areas that can transmit calls using more than one tower? b. uppose you live at (3, 5) and your friend lives at (1, 7). an you use your cell phone at either or both of your homes? c. ity is located at (21, 1) and ity is located at (4, 7). ach city has a radius of 5 miles. Which city has better coverage from the cell phone towers? 3. HOT ON You are standing at point inside a go-kart track. To determine if the track is a circle, you measure the distance to four points on the track, as shown in the diagram. What can you conclude about the shape of the track? plain. a. What is the radius of the tank? b. uppose you are standing 4 feet from another aquarium tank that has a diameter of 12 feet. How far, in feet, are you from a point of tangency? 5. XTN ON You are given seismograph readings from three locations. t (22, 3), the epicenter is 4 miles away. t (5, 21), the epicenter is 5 miles away. t (2, 5), the epicenter is 2 miles away. a. Graph circles centered at,, and with radii of 4, 5, and 2 miles, respectively. b. Locate the epicenter. c. The earthquake could be felt up to 12 miles away. If you live at (14, 16), could you feel the earthquake? plain. 6. MULTI-T OLM Use the diagram y a. Use Theorem and the quadratic formula to write an equation for y in terms of. b. ind the value of. c. ind the value of y. 706 hapter 10 roperties of ircles

60 10 ig Idea 1 HT UMMY IG I Using roperties of egments that Intersect ircles or Your Notebook You learned several relationships between tangents, secants, and chords. ome of these relationships can help you determine that two chords or tangents are congruent. or eample, tangent segments from the same eterior point are congruent. } > } Other relationships allow you to find the length of a secant or chord if you know the length of related segments. or eample, with the egments of a hord Theorem you can find the length of an unknown chord segment. p 5 p ig Idea 2 pplying ngle elationships in ircles You learned to find the measures of angles formed inside, outside, and on circles. ngles formed on circles m 5 1 } 2 m ngles formed inside circles 2 1 m } 2 1 m 1 m 2, m } 2 1 m 1 m 2 ngles formed outside circles W 3 Z X Y m } 2 1 m XY 2 m WZ 2 ig Idea 3 Using ircles in the oordinate lane y The standard equation of ( is: ( 2 h) 2 1 (y 2 k) 2 5 r 2 ( 2 2) 2 1 (y 2 1) ( 2 2) 2 1 (y 2 1) hapter ummary 707

61 10 HT VIW VIW KY VOULY classzone.com Multi-Language Glossary Vocabulary practice or a list of postulates and theorems, see pp circle, p. 651 center, radius, diameter chord, p. 651 secant, p. 651 tangent, p. 651 central angle, p. 659 minor arc, p. 659 major arc, p. 659 semicircle, p. 659 measure of a minor arc, p. 659 measure of a major arc, p. 659 congruent circles, p. 660 congruent arcs, p. 660 inscribed angle, p. 672 intercepted arc, p. 672 inscribed polygon, p. 674 circumscribed circle, p. 674 segments of a chord, p. 689 secant segment, p. 690 eternal segment, p. 690 standard equation of a circle, p. 699 VOULY XI 1. opy and complete: If a chord passes through the center of a circle, then it is called a(n)?. 2. raw and describe an inscribed angle and an intercepted arc. 3. WITING escribe how the measure of a central angle of a circle relates to the measure of the minor arc and the measure of the major arc created by the angle. In ercises 4 6, match the term with the appropriate segment. 4. Tangent segment. } LM M 5. ecant segment. } KL L 6. ternal segment. } LN K N VIW XML N XI Use the review eamples and eercises below to check your understanding of the concepts you have learned in each lesson of hapter Use roperties of Tangents pp XML In the diagram, and are points of tangency on (. ind the value of. Use Theorem 10.2 to find. 5 Tangent segments from the same point are > ubstitute olve for. 708 hapter 10 roperties of ircles

62 classzone.com hapter eview ractice XML 5 and 6 on p. 654 for s. 7 9 XI ind the value of the variable. Y and Z are points of tangency on (W. 7. W Y Z 9a a X 8. X 2c 2 1 9c 1 6 9c 1 14 Y Z W 9. X 3 9 Z r r W 10.2 ind rc Measures pp XML ind the measure of the arc of (. In the diagram, }LN is a diameter. a. MN b. NLM c. NML K N a. MN is a minor arc, so mmn 5 m MN b. NLM is a major arc, so mnlm L M c. NML is a semicircle, so mnml XML 1 and 2 on pp for s XI Use the diagram above to find the measure of the indicated arc. 10. KL 11. LM 12. KM 13. KN 10.3 pply roperties of hords pp XML In the diagram, ( > (, } > }, and m ind m. y Theorem 10.3, } and } are congruent chords in congruent circles, so the corresponding minor arcs and are congruent. o, m 5 m XML 1, 3, and 4 on pp. 664, 666 for s XI ind the measure of hapter eview 709

63 10 Use 10.4 HT VIW Inscribed ngles and olygons pp XML ind the value of each variable. LMN is inscribed in a circle, so by Theorem 10.10, opposite angles are supplementary. m L 1 m N m 1 m M a813a b a b L 3a8 b8 3a8 N 508 M a 5 30 XML 1, 2, and 5 on pp for s XI ind the value(s) of the variable(s). 17. X c8 Y 568 Z q r G 10.5 pply Other ngle elationships in ircles pp XML ind the value of y. The tangent ] Q and secant ] T intersect outside the circle, so you can use Theorem to find the value of y y y85 1 } 2 1m QT 2 m Q 2 Use Theorem T y85 1 } 2 ( ) y 5 65 ubstitute. implify. XML 2 and 3 on pp for s XI ind the value of hapter 10 roperties of ircles

64 classzone.com hapter eview ractice 10.6 ind egment Lengths in ircles pp XML ind the value of. The chords } G and } H intersect inside the circle, so you can use Theorem to find the value of. p G 5 p H Use Theorem G p p 6 ubstitute. H 5 9 olve for. XML 4 on p. 692 for. 23 XI 23. KTING INK local park has a circular ice skating rink. You are standing at point, about 12 feet from the edge of the rink. The distance from you to a point of tangency on the rink is about 20 feet. stimate the radius of the rink. r r 20 ft 12 ft 10.7 Write and Graph quations of ircles pp XML Write an equation of the circle shown. y The radius is 2 and the center is at (22, 4). ( 2 h) 2 1 (y 2 k) 2 5 r 2 tandard equation of a circle ( 2 (22)) 2 1 (y 2 4) ubstitute. ( 1 2) 2 1 (y 2 4) implify. 2 2 XML 1, 2, and 3 on pp for s XI Write an equation of the circle shown y y y Write the standard equation of the circle with the given center and radius. 27. enter (0, 0), radius enter (25, 2), radius enter (6, 21), radius enter (23, 2), radius enter (10, 7), radius enter (0, 0), radius 5.2 hapter eview 711

65 10 In HT TT (, and are points of tangency. ind the value of the variable r r Tell whether the red arcs are congruent. plain why or why not H G 2248 J K 5 L 6. M 1198 N etermine whether } is a diameter of the circle. plain your reasoning X Z Y 25 ind the indicated measure. 10. m 11. m 12. m GHJ J G H 13. m m m J 528 K 2 L M ind the value of. ound decimal answers to the nearest tenth ind the center and radius of a circle that has the standard equation ( 1 2) 2 1 (y 2 5) hapter 10 roperties of ircles

66 10 LG VIW TO INOMIL N TINOMIL lgebra classzone.com XML 1 actor using greatest common factor actor Identify the greatest common factor of the terms. The greatest common factor (G) is the product of all the common factors. irst, factor each term p p p and p 3 p p Then, write the product of the common terms. G 5 2 p p inally, use the distributive property with the G ( 1 3) XML 2 actor binomials and trinomials actor. a b olution a. Make a table of possible factorizations. ecause the middle term, 25, is negative, both factors of the third term, 3, must be negative. actors of 2 actors of 3 ossible factorization Middle term when multiplied 1, 2 23, 21 ( 2 3)(2 2 1) , 2 21, 23 ( 2 1)(2 2 3) orrect b. Use the special factoring pattern a 2 2 b 2 5 (a 1 b)(a 2 b) Write in the form a 2 2 b 2. 5 ( 1 3)( 2 3) actor using the pattern. XML 1 for s. 1 9 XML 2 for s XI actor a b 3. 9r rs t 4 1 6t t 6. 9z 3 1 3z 1 21z y 6 2 4y 5 1 2y v v v y y y 2 2 y a z 2 2 8z s 2 1 2s b b r z y 2 1 y z z lgebra eview 713

67 10 tandardized TT TION MULTIL HOI QUTION If you have difficulty solving a multiple choice question directly, you may be able to use another approach to eliminate incorrect answer choices and obtain the correct answer. OLM 1 In the diagram, nq is inscribed in a circle. The ratio of the angle measures of nq is 4:7:7. What is m Q? MTHO 1 OLV ITLY Use the Interior ngles Theorem to find m Q. Then use the fact that Q intercepts Q to find m Q. T 1 Use the ratio of the angle measures to write an equation. ecause ng is isosceles, its base angles are congruent. Let 485m Q. Then m Q 5 m You can write: m Q 1 m Q 1 m T 2 olve the equation to find the value of T 3 ind m Q. rom tep 1, m Q 5 48, so m Q 5 4 p T 4 ind m Q. ecause Q intercepts Q, m Q 5 2 p m Q. o, m Q 5 2 p The correct answer is. MTHO 2 LIMINT HOI ecause Q intercepts Q, m Q 5 } 1 p 2 m Q. lso, because nq is isosceles, its base angles, Q and, are congruent. or each choice, find m Q, m Q, and m. etermine whether the ratio of the angle measures is 4:7:7. hoice : If m Q 5 208, m Q o, m Q 1 m , and m Q 5 m 5 } The angle measures 2 108, 858, and 858 are not in the ratio 4:7:7, so hoice is not correct. hoice : If m Q 5 408, m Q o, m Q 1 m , and m Q 5 m The angle measures 208, 808, and 808 are not in the ratio 4:7:7, so hoice is not correct. hoice : If m Q 5 808, m Q o, m Q 1 m , and m Q 5 m The angle measures 408, 708, and 708 are in the ratio 4:7:7. o, m Q The correct answer is. 714 hapter 10 roperties of ircles

68 OLM 2 In the circle shown, } JK intersects } LM at point N. What is the value of? L J N K M 2 7 MTHO 1 OLV ITLY Write and solve an equation. T 1 Write an equation. y the egments of a hord Theorem, NJ p NK 5 NL p NM. You can write ( 2 2)( 2 7) 5 6 p T 2 olve the equation. ( 2 2)( 2 7) ( 2 10)( 1 1) 5 0 o, 5 10 or 521. T 3 ecide which value makes sense. If 521, then NJ ut a distance cannot be negative. If 5 10, then NJ , and NK o, The correct answer is. MTHO 2 LIMINT HOI heck to see if any choices do not make sense. T 1 heck to see if any choices give impossible values for NJ and NK. Use the fact that NJ and NK hoice : If 521, then NJ 523 and NK 528. distance cannot be negative, so you can eliminate hoice. hoice : If 5 2, then NJ 5 0 and NK 525. distance cannot be negative or 0, so you can eliminate hoice. hoice : If 5 7, then NJ 5 5 and NK 5 0. distance cannot be 0, so you can eliminate hoice. T 2 Verify that hoice is correct. y the egments of a hord Theorem, ( 2 7)( 2 2) 5 6(4). This equation is true when The correct answer is. XI plain why you can eliminate the highlighted answer choice. 1. In the diagram, what is m NQ? M N Isosceles trapezoid GH is inscribed in a circle, m 5 ( 1 8)8, and m G 5 (3 1 12)8. What is the value of? tandardized Test reparation 715

69 10 tandardized TT TI MULTIL HOI 1. In (L, } MN > } Q. Which statement is not necessarily true? MN > Q M > NQ M L N NQ > QNM MQ > NM 2. In (T, V and What is the value of? 6. In the design for a jewelry store sign, TUV is inscribed inside a circle, T 5 TU 5 12 inches, and V 5 UV 5 18 inches. What is the approimate diameter of the circle? 17 in. 25 in. T V U 22 in. 30 in. T V 7. In the diagram shown, ] Q is tangent to (N at. What is mt? N T 3. What are the coordinates of the center of a circle with equation ( 1 2) 2 1 (y 2 4) 2 5 9? (22, 24) (22, 4) (2, 24) (2, 4) 4. In the circle shown below, what is m Q? egular heagon GHJKL is inscribed in a circle. What is mkl? Two distinct circles intersect. What is the maimum number of common tangents? In the circle shown, m G and m GH What is the value of? 38 H 8 G hapter 10 roperties of ircles

70 TT TT TI classzone.com GI NW 10. } LK is tangent to (T at K. } LM is tangent to (T at M. ind the value of. T K M In (H, find m H in degrees ind the value of. 6 H 20 2 L HOT ON 13. plain why n is similar to ntq. T 14. Let 8 be the measure of an inscribed angle, and let y8 be the measure of its intercepted arc. Graph y as a function of for all possible values of. Give the slope of the graph. 15. In (J, } J > } JH. Write two true statements about congruent arcs and two true statements about congruent segments in (J. Justify each statement. J H G XTN ON 16. The diagram shows a piece of broken pottery found by an archaeologist. The archaeologist thinks that the pottery is part of a circular plate and wants to estimate the diameter of the plate. a. Trace the outermost arc of the diagram on a piece of paper. raw any two chords whose endpoints lie on the arc. b. onstruct the perpendicular bisector of each chord. Mark the point of intersection of the perpendiculars bisectors. How is this point related to the circular plate? c. ased on your results, describe a method the archaeologist could use to estimate the diameter of the actual plate. plain your reasoning. 17. The point (3, 28) lies on a circle with center (22, 4). a. Write an equation for (. b. Write an equation for the line that contains radius }. plain. c. Write an equation for the line that is tangent to ( at point. plain. tandardized Test ractice 717

Apply Other Angle Relationships in Circles

Apply Other Angle Relationships in Circles 0.5 pply Other ngle elationships in ircles efore You found the measures of angles formed on a circle. Now You will find the measures of angles inside or outside a circle. Why So you can determine the part

More information

10.4 Explore Inscribed Angles

10.4 Explore Inscribed Angles Investigating g eometry IIY se before esson 0.4 0.4 Eplore Inscribed ngles E I compass straightedge protractor Q E I O N How are inscribed angles related to central angles? he verte of a central angle

More information

Using Properties of Segments that Intersect Circles

Using Properties of Segments that Intersect Circles ig Idea 1 H UY I I Using roperties of egments that Intersect ircles or Your otebook You learned several relationships between tangents, secants, and chords. ome of these relationships can help you determine

More information

10.6 Investigate Segment Lengths

10.6 Investigate Segment Lengths Investigating g Geometry TIVITY. Investigate Segment Lengths M T R I LS graphing calculator or computer Use before Lesson. classzone.com Keystrokes Q U S T I O N What is the relationship between the lengths

More information

10-1 Study Guide and Intervention

10-1 Study Guide and Intervention opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-1 tudy Guide and Intervention ircles and ircumference arts of ircles circle consists of all points in a plane that are

More information

10.2. Find Arc Measures. For Your Notebook. } RT is a diameter, so C RST is a semicircle, and m C RST Find measures of arcs KEY CONCEPT

10.2. Find Arc Measures. For Your Notebook. } RT is a diameter, so C RST is a semicircle, and m C RST Find measures of arcs KEY CONCEPT 10.2 Find rc Measures efore ou found angle measures. Now ou will use angle measures to find arc measures. Why? o you can describe the arc made by a bridge, as in Ex. 22. Key Vocabulary central angle minor

More information

Use Properties of Tangents

Use Properties of Tangents roperties of ircles 1010.1 Use roperties of Tangents 10.2 ind rc Measures 10.3 ppl roperties of hords 10.4 Use Inscribed ngles and olgons 10.5 ppl Other ngle elationships in ircles 10.6 ind egment Lengths

More information

8.2 Investigate Parallelograms

8.2 Investigate Parallelograms Investigating g Geometry TIVITY 8.2 Investigate arallelograms T E I graphing calculator or computer Use before esson 8.2 classzone.com Keystrokes Q U E T I O N What are some of the properties of a parallelogram?

More information

Chords and Arcs. Objectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords

Chords and Arcs. Objectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords - hords and rcs ommon ore State Standards G-.. Identify and describe relationships among inscribed angles, radii, and chords. M, M bjectives To use congruent chords, arcs, and central angles To use perpendicular

More information

Geometry: A Complete Course

Geometry: A Complete Course eometry: omplete ourse with rigonometry) odule - tudent Worket Written by: homas. lark Larry. ollins 4/2010 or ercises 20 22, use the diagram below. 20. ssume is a rectangle. a) f is 6, find. b) f is,

More information

Using Chords. Essential Question What are two ways to determine when a chord is a diameter of a circle?

Using Chords. Essential Question What are two ways to determine when a chord is a diameter of a circle? 10.3 Using hords ssential uestion What are two ways to determine when a chord is a diameter of a circle? rawing iameters OOKI O UU o be proficient in math, you need to look closely to discern a pattern

More information

Ready To Go On? Skills Intervention 11-1 Lines That Intersect Circles

Ready To Go On? Skills Intervention 11-1 Lines That Intersect Circles Name ate lass STION 11 Ready To Go On? Skills Intervention 11-1 Lines That Intersect ircles ind these vocabulary words in Lesson 11-1 and the Multilingual Glossary. Vocabulary interior of a circle exterior

More information

UNIT OBJECTIVES. unit 9 CIRCLES 259

UNIT OBJECTIVES. unit 9 CIRCLES 259 UNIT 9 ircles Look around whatever room you are in and notice all the circular shapes. Perhaps you see a clock with a circular face, the rim of a cup or glass, or the top of a fishbowl. ircles have perfect

More information

6.6 Investigate Proportionality

6.6 Investigate Proportionality Investigating g Geometry TIVITY 6.6 Investigate roportionality M T I LS graphing calculator or computer Use before Lesson 6.6 classzone.com Keystrokes Q U S T I O N How can you use geometry drawing software

More information

Objectives To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord

Objectives To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord 1-3 Inscribed ngles ommon ore State Standards G-.. Identify and describe relationships among inscribed angles, radii, and chords. lso G-..3, G-..4 M 1, M 3, M 4, M 6 bjectives To find the measure of an

More information

15.3 Tangents and Circumscribed Angles

15.3 Tangents and Circumscribed Angles Name lass ate 15.3 Tangents and ircumscribed ngles Essential uestion: What are the key theorems about tangents to a circle? esource Locker Explore Investigating the Tangent-adius Theorem tangent is a line

More information

14.3 Tangents and Circumscribed Angles

14.3 Tangents and Circumscribed Angles Name lass Date 14.3 Tangents and ircumscribed ngles Essential uestion: What are the key theorems about tangents to a circle? Explore G.5. Investigate patterns to make conjectures about geometric relationships,

More information

Circles and Arcs. Objectives To find the measures of central angles and arcs To find the circumference and arc length

Circles and Arcs. Objectives To find the measures of central angles and arcs To find the circumference and arc length 10-6 ircles and rcs ommon ore tate tandards G-..1 Know precise definitions of... circle... G-..1 rove that all circles are similar. lso G-..2, G-..5 M 1, M 3, M 4, M 6, M 8 bjectives o find the measures

More information

radii: AP, PR, PB diameter: AB chords: AB, CD, AF secant: AG or AG tangent: semicircles: ACB, ARB minor arcs: AC, AR, RD, BC,

radii: AP, PR, PB diameter: AB chords: AB, CD, AF secant: AG or AG tangent: semicircles: ACB, ARB minor arcs: AC, AR, RD, BC, h 6 Note Sheets L Shortened Key Note Sheets hapter 6: iscovering and roving ircle roperties eview: ircles Vocabulary If you are having problems recalling the vocabulary, look back at your notes for Lesson

More information

Skills Practice Skills Practice for Lesson 11.1

Skills Practice Skills Practice for Lesson 11.1 Skills Practice Skills Practice for Lesson.1 Name ate Riding a Ferris Wheel Introduction to ircles Vocabulary Identify an instance of each term in the diagram. 1. circle X T 2. center of the circle H I

More information

Riding a Ferris Wheel

Riding a Ferris Wheel Lesson.1 Skills Practice Name ate iding a Ferris Wheel Introduction to ircles Vocabulary Identify an instance of each term in the diagram. 1. center of the circle 6. central angle T H I 2. chord 7. inscribed

More information

Proportions in Triangles

Proportions in Triangles - roportions in Triangles ontent tandard G.RT. rove theorems about triangles...a line parallel to one side of a triangle divides the other two proportionally... Objective To use the ide-plitter Theorem

More information

Solve problems involving tangents to a circle. Solve problems involving chords of a circle

Solve problems involving tangents to a circle. Solve problems involving chords of a circle 8UNIT ircle Geometry What You ll Learn How to Solve problems involving tangents to a circle Solve problems involving chords of a circle Solve problems involving the measures of angles in a circle Why Is

More information

Introduction Circle Some terms related with a circle

Introduction Circle Some terms related with a circle 141 ircle Introduction In our day-to-day life, we come across many objects which are round in shape, such as dials of many clocks, wheels of a vehicle, bangles, key rings, coins of denomination ` 1, `

More information

b. Find the measures of the two angles formed by the chord and the tangent line.

b. Find the measures of the two angles formed by the chord and the tangent line. 0.5 NI NOW N I.5... ngle Relationships in ircles ssential Question When a chord intersects a tangent line or another chord, what relationships exist aong the angles and arcs fored? ngles ored by a hord

More information

Chapter 1. Some Basic Theorems. 1.1 The Pythagorean Theorem

Chapter 1. Some Basic Theorems. 1.1 The Pythagorean Theorem hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a 2 + b 2 = c 2. roof. b a a 3 2 b 2 b 4 b a b

More information

Circles-Tangent Properties

Circles-Tangent Properties 15 ircles-tangent roperties onstruction of tangent at a point on the circle. onstruction of tangents when the angle between radii is given. Tangents from an external point - construction and proof Touching

More information

Theorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C.

Theorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C. hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a + b = c. roof. b a a 3 b b 4 b a b 4 1 a a 3

More information

and Congruence You learned about points, lines, and planes. You will use segment postulates to identify congruent segments.

and Congruence You learned about points, lines, and planes. You will use segment postulates to identify congruent segments. 1.2 Use Segments and ongruence efore Now You learned about points, lines, and planes. You will use segment postulates to identify congruent segments. Why? So you can calculate flight distances, as in x.

More information

Work with a partner. Use dynamic geometry software to draw any ABC. a. Bisect B and plot point D at the intersection of the angle bisector and AC.

Work with a partner. Use dynamic geometry software to draw any ABC. a. Bisect B and plot point D at the intersection of the angle bisector and AC. .6 Proportionality heorems ssential uestion hat proportionality relationships eist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? iscovering a Proportionality

More information

Seismograph (p. 582) Car (p. 554) Dartboard (p. 547) Bicycle Chain (p. 539)

Seismograph (p. 582) Car (p. 554) Dartboard (p. 547) Bicycle Chain (p. 539) 10 ircles 10.1 ines and egments hat Intersect ircles 10. inding rc easures 10.3 Using hords 10.4 Inscribed ngles and olygons 10.5 ngle elationships in ircles 10.6 egment elationships in ircles 10.7 ircles

More information

If the measure ofaacb is less than 180, then A, B, and all the points on C that lie in the

If the measure ofaacb is less than 180, then A, B, and all the points on C that lie in the age 1 of 7 11.3 rcs and entral ngles oal Use properties of arcs of circles. Key Words minor arc major arc semicircle congruent circles congruent arcs arc length ny two points and on a circle determine

More information

What You ll Learn. Why It s Important. We see circles in nature and in design. What do you already know about circles?

What You ll Learn. Why It s Important. We see circles in nature and in design. What do you already know about circles? We see circles in nature and in design. What do you already know about circles? What You ll Learn ircle properties that relate: a tangent to a circle and the radius of the circle a chord in a circle, its

More information

New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative

New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative Slide 1 / 150 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students

More information

Practice For use with pages

Practice For use with pages Name ate ON 0. ractice For use with pages 678 686 se ( to draw the described part of the circle.. raw a diameter and label it }.. raw a tangent ra and label it ###$. 3. raw a secant and label it } F. 4.

More information

Study Guide. Exploring Circles. Example: Refer to S for Exercises 1 6.

Study Guide. Exploring Circles. Example: Refer to S for Exercises 1 6. 9 1 Eploring ircles A circle is the set of all points in a plane that are a given distance from a given point in the plane called the center. Various parts of a circle are labeled in the figure at the

More information

Chapter 19 Exercise 19.1

Chapter 19 Exercise 19.1 hapter 9 xercise 9... (i) n axiom is a statement that is accepted but cannot be proven, e.g. x + 0 = x. (ii) statement that can be proven logically: for example, ythagoras Theorem. (iii) The logical steps

More information

Geo - CH11 Practice Test

Geo - CH11 Practice Test Geo - H11 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Identify the secant that intersects ñ. a. c. b. l d. 2. satellite rotates 50 miles

More information

What is the longest chord?.

What is the longest chord?. Section: 7-6 Topic: ircles and rcs Standard: 7 & 21 ircle Naming a ircle Name: lass: Geometry 1 Period: Date: In a plane, a circle is equidistant from a given point called the. circle is named by its.

More information

1.2 Perpendicular Lines

1.2 Perpendicular Lines Name lass ate 1.2 erpendicular Lines Essential Question: What are the key ideas about perpendicular bisectors of a segment? 1 Explore onstructing erpendicular isectors and erpendicular Lines You can construct

More information

NAME DATE PERIOD. 4. If m ABC x and m BAC m BCA 2x 10, is B F an altitude? Explain. 7. Find x if EH 16 and FH 6x 5. G

NAME DATE PERIOD. 4. If m ABC x and m BAC m BCA 2x 10, is B F an altitude? Explain. 7. Find x if EH 16 and FH 6x 5. G 5- NM IO ractice isectors, Medians, and ltitudes LG In, is the angle bisector of,,, and are medians, and is the centroid.. ind x if 4x and 0.. ind y if y and 8.. ind z if 5z 0 and 4. 4. If m x and m m

More information

Using Properties of Special Segments in Triangles. Using Triangle Inequalities to Determine What Triangles are Possible

Using Properties of Special Segments in Triangles. Using Triangle Inequalities to Determine What Triangles are Possible 5 ig Idea 1 HTR SUMMRY IG IS Using roperties of Special Segments in Triangles For Your otebook Special segment Midsegment erpendicular bisector ngle bisector Median (connects verte to midpoint of opposite

More information

4. 2 common tangents 5. 1 common tangent common tangents 7. CE 2 0 CD 2 1 DE 2

4. 2 common tangents 5. 1 common tangent common tangents 7. CE 2 0 CD 2 1 DE 2 hapter 0 opright b Mcougal Littell, a division of Houghton Mifflin ompan. Prerequisite Skills (p. 648). Two similar triangles have congruent corresponding angles and proportional corresponding sides..

More information

Use Properties of Tangents

Use Properties of Tangents opeties of icles 1010.1 Use opeties of Tangents 10.2 Find c Measues 10.3 pply opeties of hods 10.4 Use Inscibed ngles and olygons 10.5 pply Othe ngle elationships in icles 10.6 Find egment Lengths in icles

More information

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle 10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by

More information

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane?

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane? GMTRY: XM () Name: 1. How many non collinear points determine a plane? ) none ) one ) two ) three 2. How many edges does a heagonal prism have? ) 6 ) 12 ) 18 ) 2. Name the intersection of planes Q and

More information

10.3 Start Thinking Warm Up Cumulative Review Warm Up

10.3 Start Thinking Warm Up Cumulative Review Warm Up 10.3 tart hinking etermine if the statement is always true, sometimes true, or never true. plain your reasoning. 1. chord is a diameter. 2. diameter is a chord. 3. chord and a radius have the same measure.

More information

Using the Pythagorean Theorem and Its Converse

Using the Pythagorean Theorem and Its Converse 7 ig Idea 1 HPTR SUMMR IG IDS Using the Pythagorean Theorem and Its onverse For our Notebook The Pythagorean Theorem states that in a right triangle the square of the length of the hypotenuse c is equal

More information

Replacement for a Carpenter s Square

Replacement for a Carpenter s Square Lesson.1 Skills Practice Name Date Replacement for a arpenter s Square Inscribed and ircumscribed Triangles and Quadrilaterals Vocabulary nswer each question. 1. How are inscribed polygons and circumscribed

More information

Mth 076: Applied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE

Mth 076: Applied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE Mth 076: pplied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE INTRODUTION TO GEOMETRY Pick up Geometric Formula Sheet (This sheet may be used while testing) ssignment Eleven: Problems Involving

More information

Final Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Final Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question. ( Final Exam Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. is an isosceles triangle. is the longest side with length. = and =. Find. 4 x + 4 7

More information

Reteaching , or 37.5% 360. Geometric Probability. Name Date Class

Reteaching , or 37.5% 360. Geometric Probability. Name Date Class Name ate lass Reteaching Geometric Probability INV 6 You have calculated probabilities of events that occur when coins are tossed and number cubes are rolled. Now you will learn about geometric probability.

More information

Chapter 6. Worked-Out Solutions. Chapter 6 Maintaining Mathematical Proficiency (p. 299)

Chapter 6. Worked-Out Solutions. Chapter 6 Maintaining Mathematical Proficiency (p. 299) hapter 6 hapter 6 Maintaining Mathematical Proficiency (p. 99) 1. Slope perpendicular to y = 1 x 5 is. y = x + b 1 = + b 1 = 9 + b 10 = b n equation of the line is y = x + 10.. Slope perpendicular to y

More information

Unit 10 Geometry Circles. NAME Period

Unit 10 Geometry Circles. NAME Period Unit 10 Geometry Circles NAME Period 1 Geometry Chapter 10 Circles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (10-1) Circles and Circumference

More information

Common Core Readiness Assessment 4

Common Core Readiness Assessment 4 ommon ore Readiness ssessment 4 1. Use the diagram and the information given to complete the missing element of the two-column proof. 2. Use the diagram and the information given to complete the missing

More information

SM2H Unit 6 Circle Notes

SM2H Unit 6 Circle Notes Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:

More information

Circles. II. Radius - a segment with one endpoint the center of a circle and the other endpoint on the circle.

Circles. II. Radius - a segment with one endpoint the center of a circle and the other endpoint on the circle. Circles Circles and Basic Terminology I. Circle - the set of all points in a plane that are a given distance from a given point (called the center) in the plane. Circles are named by their center. II.

More information

( ) Chapter 10 Review Question Answers. Find the value of x mhg. m B = 1 2 ( 80 - x) H x G. E 30 = 80 - x. x = 50. Find m AXB and m Y A D X 56

( ) Chapter 10 Review Question Answers. Find the value of x mhg. m B = 1 2 ( 80 - x) H x G. E 30 = 80 - x. x = 50. Find m AXB and m Y A D X 56 hapter 10 Review Question nswers 1. ( ) Find the value of mhg 30 m = 1 2 ( 30) = 15 F 80 m = 1 2 ( 80 - ) H G E 30 = 80 - = 50 2. Find m X and m Y m X = 1 120 + 56 2 ( ) = 88 120 X 56 Y m Y = 1 120-56

More information

Circles and Volume. Circle Theorems. Essential Questions. Module Minute. Key Words. What To Expect. Analytical Geometry Circles and Volume

Circles and Volume. Circle Theorems. Essential Questions. Module Minute. Key Words. What To Expect. Analytical Geometry Circles and Volume Analytical Geometry Circles and Volume Circles and Volume There is something so special about a circle. It is a very efficient shape. There is no beginning, no end. Every point on the edge is the same

More information

Incoming Magnet Precalculus / Functions Summer Review Assignment

Incoming Magnet Precalculus / Functions Summer Review Assignment Incoming Magnet recalculus / Functions Summer Review ssignment Students, This assignment should serve as a review of the lgebra and Geometry skills necessary for success in recalculus. These skills were

More information

Riding a Ferris Wheel. Students should be able to answer these questions after Lesson 10.1:

Riding a Ferris Wheel. Students should be able to answer these questions after Lesson 10.1: .1 Riding a Ferris Wheel Introduction to ircles Students should be able to answer these questions after Lesson.1: What are the parts of a circle? How are the parts of a circle drawn? Read Question 1 and

More information

Chapter-wise questions

Chapter-wise questions hapter-wise questions ircles 1. In the given figure, is circumscribing a circle. ind the length of. 3 15cm 5 2. In the given figure, is the center and. ind the radius of the circle if = 18 cm and = 3cm

More information

Name Date Period. Notes - Tangents. 1. If a line is a tangent to a circle, then it is to the

Name Date Period. Notes - Tangents. 1. If a line is a tangent to a circle, then it is to the Name ate Period Notes - Tangents efinition: tangent is a line in the plane of a circle that intersects the circle in eactly one point. There are 3 Theorems for Tangents. 1. If a line is a tangent to a

More information

1. Draw and label a diagram to illustrate the property of a tangent to a circle.

1. Draw and label a diagram to illustrate the property of a tangent to a circle. Master 8.17 Extra Practice 1 Lesson 8.1 Properties of Tangents to a Circle 1. Draw and label a diagram to illustrate the property of a tangent to a circle. 2. Point O is the centre of the circle. Points

More information

Circles. Exercise 9.1

Circles. Exercise 9.1 9 uestion. Exercise 9. How many tangents can a circle have? Solution For every point of a circle, we can draw a tangent. Therefore, infinite tangents can be drawn. uestion. Fill in the blanks. (i) tangent

More information

Chapter 10. Properties of Circles

Chapter 10. Properties of Circles Chapter 10 Properties of Circles 10.1 Use Properties of Tangents Objective: Use properties of a tangent to a circle. Essential Question: how can you verify that a segment is tangent to a circle? Terminology:

More information

MF9SB_CH09_p pp8.qxd 4/15/09 6:51 AM Page NEL

MF9SB_CH09_p pp8.qxd 4/15/09 6:51 AM Page NEL 408 NEL hapter 9 ircle Geometry GLS ou will be able to identify and apply the relationships between inscribed angles and central angles use properties of chords to solve problems identify properties of

More information

Geometry. Chapter 10 Resource Masters

Geometry. Chapter 10 Resource Masters Geometry hapter 10 esource Masters 10 eading to Learn Mathematics Vocabulary uilder This is an alphabetical list of the key vocabulary terms you will learn in hapter 10. s you study the chapter, complete

More information

0809ge. Geometry Regents Exam Based on the diagram below, which statement is true?

0809ge. Geometry Regents Exam Based on the diagram below, which statement is true? 0809ge 1 Based on the diagram below, which statement is true? 3 In the diagram of ABC below, AB AC. The measure of B is 40. 1) a b ) a c 3) b c 4) d e What is the measure of A? 1) 40 ) 50 3) 70 4) 100

More information

Geometry Final Review. Chapter 1. Name: Per: Vocab. Example Problems

Geometry Final Review. Chapter 1. Name: Per: Vocab. Example Problems Geometry Final Review Name: Per: Vocab Word Acute angle Adjacent angles Angle bisector Collinear Line Linear pair Midpoint Obtuse angle Plane Pythagorean theorem Ray Right angle Supplementary angles Complementary

More information

Assignment. Riding a Ferris Wheel Introduction to Circles. 1. For each term, name all of the components of circle Y that are examples of the term.

Assignment. Riding a Ferris Wheel Introduction to Circles. 1. For each term, name all of the components of circle Y that are examples of the term. ssignment ssignment for Lesson.1 Name Date Riding a Ferris Wheel Introduction to ircles 1. For each term, name all of the components of circle Y that are examples of the term. G R Y O T M a. hord GM, R,

More information

Geometry Honors Homework

Geometry Honors Homework Geometry Honors Homework pg. 1 12-1 Practice Form G Tangent Lines Algebra Assume that lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? 1. 2. 3. The circle

More information

Answers. Chapter10 A Start Thinking. and 4 2. Sample answer: no; It does not pass through the center.

Answers. Chapter10 A Start Thinking. and 4 2. Sample answer: no; It does not pass through the center. hapter10 10.1 Start Thinking 6. no; is not a right triangle because the side lengths do not satisf the Pthagorean Theorem (Thm. 9.1). 1. (3, ) 7. es; is a right triangle because the side lengths satisf

More information

Graph and Write Equations of Circles

Graph and Write Equations of Circles TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges,

More information

Plane geometry Circles: Problems with some Solutions

Plane geometry Circles: Problems with some Solutions The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the

More information

Indicate whether the statement is true or false.

Indicate whether the statement is true or false. PRACTICE EXAM IV Sections 6.1, 6.2, 8.1 8.4 Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number.

More information

Lesson 1.7 circles.notebook. September 19, Geometry Agenda:

Lesson 1.7 circles.notebook. September 19, Geometry Agenda: Geometry genda: Warm-up 1.6(need to print of and make a word document) ircle Notes 1.7 Take Quiz if you were not in class on Friday Remember we are on 1.7 p.72 not lesson 1.8 1 Warm up 1.6 For Exercises

More information

0114ge. Geometry Regents Exam 0114

0114ge. Geometry Regents Exam 0114 0114ge 1 The midpoint of AB is M(4, 2). If the coordinates of A are (6, 4), what are the coordinates of B? 1) (1, 3) 2) (2, 8) 3) (5, 1) 4) (14, 0) 2 Which diagram shows the construction of a 45 angle?

More information

Example 1: Finding angle measures: I ll do one: We ll do one together: You try one: ML and MN are tangent to circle O. Find the value of x

Example 1: Finding angle measures: I ll do one: We ll do one together: You try one: ML and MN are tangent to circle O. Find the value of x Ch 1: Circles 1 1 Tangent Lines 1 Chords and Arcs 1 3 Inscribed Angles 1 4 Angle Measures and Segment Lengths 1 5 Circles in the coordinate plane 1 1 Tangent Lines Focused Learning Target: I will be able

More information

0611ge. Geometry Regents Exam Line segment AB is shown in the diagram below.

0611ge. Geometry Regents Exam Line segment AB is shown in the diagram below. 0611ge 1 Line segment AB is shown in the diagram below. In the diagram below, A B C is a transformation of ABC, and A B C is a transformation of A B C. Which two sets of construction marks, labeled I,

More information

Evaluate: Homework and Practice

Evaluate: Homework and Practice valuate: Homework and ractice Use the figure for ercises 1 2. Suppose ou use geometr software to construct two chords S and TU that intersect inside a circle at V. Online Homework Hints and Help tra ractice

More information

Use Properties of Tangents

Use Properties of Tangents 6.1 Georgia Performance Standard(s) MM2G3a, MM2G3d Your Notes Use Properties of Tangents Goal p Use properties of a tangent to a circle. VOULRY ircle enter Radius hord iameter Secant Tangent Example 1

More information

Lesson 9.1 Skills Practice

Lesson 9.1 Skills Practice Lesson 9.1 Skills Practice Name Date Earth Measure Introduction to Geometry and Geometric Constructions Vocabulary Write the term that best completes the statement. 1. means to have the same size, shape,

More information

Park Forest Math Team. Meet #4. Geometry. Self-study Packet

Park Forest Math Team. Meet #4. Geometry. Self-study Packet Park Forest Math Team Meet #4 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. : ngle measures in plane figures including supplements and complements 3. Number Theory:

More information

Click on a topic to go to that section. Euclid defined a circle and its center in this way: Euclid defined figures in this way:

Click on a topic to go to that section. Euclid defined a circle and its center in this way: Euclid defined figures in this way: lide 1 / 59 lide / 59 New Jersey enter for eaching and Learning Progressive Mathematics Initiative his material is made freely available at www.njctl.org and is intended for the non-commercial use of students

More information

Which statement is true about parallelogram FGHJ and parallelogram F ''G''H''J ''?

Which statement is true about parallelogram FGHJ and parallelogram F ''G''H''J ''? Unit 2 Review 1. Parallelogram FGHJ was translated 3 units down to form parallelogram F 'G'H'J '. Parallelogram F 'G'H'J ' was then rotated 90 counterclockwise about point G' to obtain parallelogram F

More information

National Benchmark Test 1. 1 Which three-dimensional figure does this net produce? Name: Date: Copyright by Pearson Education Page 1 of 13

National Benchmark Test 1. 1 Which three-dimensional figure does this net produce? Name: Date: Copyright by Pearson Education Page 1 of 13 National enchmark Test 1 Name: ate: 1 Which three-dimensional figure does this net produce? opyright 2005-2006 by Pearson Education Page 1 of 13 National enchmark Test 1 2 Which of the following is a net

More information

Geometry Honors Final Exam Review June 2018

Geometry Honors Final Exam Review June 2018 Geometry Honors Final Exam Review June 2018 1. Determine whether 128 feet, 136 feet, and 245 feet can be the lengths of the sides of a triangle. 2. Casey has a 13-inch television and a 52-inch television

More information

11. Concentric Circles: Circles that lie in the same plane and have the same center.

11. Concentric Circles: Circles that lie in the same plane and have the same center. Circles Definitions KNOW THESE TERMS 1. Circle: The set of all coplanar points equidistant from a given point. 2. Sphere: The set of all points equidistant from a given point. 3. Radius of a circle: The

More information

Geometry. Chapter 10 Resource Masters

Geometry. Chapter 10 Resource Masters Geometry hapter 10 esource Masters onsumable Workbooks Many of the worksheets contained in the hapter esource Masters booklets are available as consumable workbooks. tudy Guide and Intervention Workbook

More information

So, PQ is about 3.32 units long Arcs and Chords. ALGEBRA Find the value of x.

So, PQ is about 3.32 units long Arcs and Chords. ALGEBRA Find the value of x. ALGEBRA Find the value of x. 1. Arc ST is a minor arc, so m(arc ST) is equal to the measure of its related central angle or 93. and are congruent chords, so the corresponding arcs RS and ST are congruent.

More information

NAME DATE PERIOD. Study Guide and Intervention

NAME DATE PERIOD. Study Guide and Intervention 7-1 tudy Guide and Intervention atios and Proportions Write and Use atios ratio is a comparison of two quantities by divisions. The ratio a to b, where b is not zero, can be written as a b or a:b. ample

More information

UNIT 5 SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF Unit Assessment

UNIT 5 SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF Unit Assessment Unit 5 ircle the letter of the best answer. 1. line segment has endpoints at (, 5) and (, 11). point on the segment has a distance that is 1 of the length of the segment from endpoint (, 5). What are the

More information

2013 ACTM Regional Geometry Exam

2013 ACTM Regional Geometry Exam 2013 TM Regional Geometry Exam In each of the following choose the EST answer and record your choice on the answer sheet provided. To insure correct scoring, be sure to make all erasures completely. The

More information

( ) ( ) Geometry Team Solutions FAMAT Regional February = 5. = 24p.

( ) ( ) Geometry Team Solutions FAMAT Regional February = 5. = 24p. . A 6 6 The semi perimeter is so the perimeter is 6. The third side of the triangle is 7. Using Heron s formula to find the area ( )( )( ) 4 6 = 6 6. 5. B Draw the altitude from Q to RP. This forms a 454590

More information

Chapter 12 Practice Test

Chapter 12 Practice Test hapter 12 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. ssume that lines that appear to be tangent are tangent. is the center of the circle.

More information

GEOMETRY REVIEW FOR MIDTERM

GEOMETRY REVIEW FOR MIDTERM Y VIW I he midterm eam for period is on /, 0:00 to :. he eam will consist of approimatel 0 multiple-choice and open-ended questions. Now is the time to start studing!!! PP eviews all previous assessments.

More information

How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots

How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots . Approximating Square Roots How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots Work with a partner. Archimedes was a Greek mathematician,

More information

Given that m A = 50 and m B = 100, what is m Z? A. 15 B. 25 C. 30 D. 50

Given that m A = 50 and m B = 100, what is m Z? A. 15 B. 25 C. 30 D. 50 UNIT : SIMILARITY, CONGRUENCE AND PROOFS ) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of. The dilation is centered at ( 4, ). ) Which transformation results in a figure that is similar

More information

Correlation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1

Correlation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1 Correlation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1 ALGEBRA I A.1 Mathematical process standards. The student

More information