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 in the oordinate lane the ig Idea eismograph (p. 58) aturn (p. 577) 7) ar (p. 554) artboard (p. 547) icycle hain (p. 539) athematical hinking: athematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
aintaining athematical roficiency ultiplying inomials (.10.) ample 1 ind the product ( + 3)( 1). irst Outer Inner ast ( + 3)( 1) = () + ( 1) + 3() + (3)( 1) OI ethod he product is + 5 3. = + ( ) + 6 + ( 3) ultiply. = + 5 3 implify. ind the product. 1. ( + 7)( + 4). (a + 1)(a 5) 3. (q 9)(3q 4) 4. (v 7)(5v + 1) 5. (4h + 3)( + h) 6. (8 6b)(5 3b) olving uadratic quations by ompleting the quare (.8.) ample olve + 8 3 = 0 by completing the square. + 8 3 = 0 + 8 = 3 Write original equation. dd 3 to each side. + 8 + 4 = 3 + 4 omplete the square by adding ( 8 ), or 4, to each side. ( + 4) = 19 Write the left side as a square of a binomial. + 4 = ± 19 = 4 ± 19 ake the square root of each side. ubtract 4 from each side. he solutions are = 4 + 19 0.36 and = 4 19 8.36. olve the equation by completing the square. ound your answer to the nearest hundredth, if necessary. 7. = 5 8. r + 10r = 7 9. w 8w = 9 10. p + 10p 4 = 0 11. k 4k 7 = 0 1. z + z = 1 13. ONIN Write an epression that represents the product of two consecutive positive odd integers. plain your reasoning. 531
athematical hinking nalyzing elationships of ircles ore oncept ircles and angent ircles 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 as. athematically profi cient students use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. (.1.) circle, or oplanar circles that intersect in one point are called tangent circles. and are tangent circles. and are tangent circles. elationships of ircles and angent ircles a. ach circle at the right consists of points that are 3 units from the center. What is the greatest distance from any point on to any point on? b. hree circles,,, and, consist of points that are 3 units from their centers. he centers,, and of the circles are collinear, is tangent to, and is tangent to. What is the distance from to? OUION a. ecause the points on each circle are 3 units from the center, the greatest distance from any point on to any point on is 3 + 3 + 3 = 9 units. b. ecause,, and are collinear, is tangent 3 3 3 to, and is tangent to, the circles are as shown. o, the distance from to is 3 + 3 = 6 units. 3 3 onitoring rogress et,, and consist of points that are 3 units from the centers. 1. raw so that it passes through points and in the figure at the right. plain your reasoning.. raw,, and so that each is tangent to the other two. raw a larger circle,, that is tangent to each of the other three circles. Is the distance from point to a point on less than, greater than, or equal to 6? plain. 53 hapter 10 ircles
10.1 X NI KNOW N KI.5..1. ines and egments hat Intersect ircles ssential uestion What are the definitions of the lines and segments that intersect a circle? ines and ine egments hat Intersect ircles Work with a partner. he drawing at the right shows five lines or segments that intersect a circle. Use the relationships shown to write a definition for each type of line or segment. hen use the Internet or some other resource to verify your definitions. hord: ecant: chord diameter tangent radius angent: adius: secant iameter: Using tring to raw a ircle Work with a partner. Use two pencils, a piece of string, and a piece of paper. a. ie the two ends of the piece of string loosely around the two pencils. b. nchor one pencil on the paper at the center of the circle. Use the other pencil to draw a circle around the anchor point while using slight pressure to keep the string taut. o not let the string wind around either pencil. ONIN o be proficient in math, you need to know and fleibly use different properties of operations and objects. c. plain how the distance between the two pencil points as you draw the circle is related to two of the lines or line segments you defined in ploration 1. ommunicate Your nswer 3. What are the definitions of the lines and segments that intersect a circle? 4. Of the five types of lines and segments in ploration 1, which one is a subset of another? plain. 5. plain how to draw a circle with a diameter of 8 inches. ection 10.1 ines and egments hat Intersect ircles 533
10.1 esson What You Will earn ore Vocabulary circle, p. 534 center, p. 534 radius, p. 534 chord, p. 534 diameter, p. 534 secant, p. 534 tangent, p. 534 point of tangency, p. 534 tangent circles, p. 535 concentric circles, p. 535 common tangent, p. 535 IN he words radius and diameter refer to 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. Identify special segments and lines. raw and identify common tangents. Use properties of tangents. Identifying pecial egments and ines 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 as. ore oncept ines and egments hat Intersect ircles segment whose endpoints are the center and any point on a 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. 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. he tangent ray and the tangent segment are also called tangents. chord center diameter secant point of tangency tangent circle, or radius Identifying pecial egments and ines UY I In this book, assume that all segments, rays, or lines that appear to be tangent to a circle are tangents. OUION ell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of. a. b. c. d. 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 in eactly one point. d. is a secant because it is a line that intersects the circle in two points. onitoring rogress Help in nglish and panish at igideasath.com 1. In ample 1, what word best describes??. In ample 1, name a tangent and a tangent segment. 534 hapter 10 ircles
rawing and Identifying ommon angents ore oncept oplanar ircles and ommon angents In a plane, 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 circles. points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles line or segment that is tangent to two coplanar circles is called a common tangent. common internal tangent intersects the segment that joins the centers of the two circles. common eternal tangent does not intersect the segment that joins the centers of the two circles. rawing and Identifying ommon angents ell how many common tangents the circles have and draw them. Use blue to indicate common eternal tangents and red to indicate common internal tangents. a. b. c. OUION raw the segment that joins the centers of the two circles. hen draw the common tangents. Use blue to indicate lines that do not intersect the segment joining the centers and red to indicate lines that intersect the segment joining the centers. a. 4 common tangents b. 3 common tangents c. common tangents onitoring rogress Help in nglish and panish at igideasath.com ell how many common tangents the circles have and draw them. tate whether the tangents are eternal tangents or internal tangents. 3. 4. 5. ection 10.1 ines and egments hat Intersect ircles 535
Using roperties of angents heorems heorem 10.1 angent ine to ircle heorem 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. 47, p. 540 ine m is tangent to if and only if m. heorem 10. ternal angent ongruence heorem angent segments from a common eternal point are congruent. roof. 46, p. 540 If and are tangent segments, then. Is tangent to? OUION Verifying a angent to a ircle 35 37 Use the onverse of the ythagorean heorem (heorem 9.). ecause 1 + 35 = 37, is a right triangle and. o, is perpendicular to a radius of at its endpoint on. y the angent ine to ircle heorem, is tangent to. inding the adius of a ircle 1 In the diagram, point is a point of tangency. ind the radius r of. 50 ft r 80 ft r OUION You know from the angent ine to ircle heorem that, so is a right triangle. You can use the ythagorean heorem (heorem 9.1). = + ythagorean heorem (r + 50) = r + 80 ubstitute. r + 100r + 500 = r + 6400 ultiply. 100r = 3900 ubtract r and 500 from each side. r = 39 ivide each side by 100. he radius is 39 feet. 536 hapter 10 ircles
onstructing a angent to a ircle iven and point, construct a line tangent to that passes through. Use a compass and straightedge. OUION tep 1 tep tep 3 ind a midpoint raw. onstruct the bisector of the segment and label the midpoint. raw a circle onstruct with radius. abel one of the points where intersects as point. onstruct a tangent line raw. It is a tangent to that passes through. Using roperties of angents is tangent to at, and is tangent to at. ind the value of. 8 3 + 4 OUION = 8 = 3 + 4 ubstitute. 8 = olve for. he value of is 8. onitoring rogress 6. Is tangent to? 3 4 ternal angent ongruence heorem Help in nglish and panish at igideasath.com 7. is tangent to. ind the radius of. r r 4 18 8. oints and N are points of tangency. ind the value(s) of. N 9 ection 10.1 ines and egments hat Intersect ircles 537
10.1 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. WIIN How are chords and secants alike? How are they different?. WIIN plain how you can determine from the contet whether the words radius and diameter are referring to segments or lengths. 3. O H NN oplanar circles that have a common center are called. 4. WHIH ON ON ON? Which segment does not belong with the other three? plain your reasoning. chord radius tangent diameter onitoring rogress and odeling with athematics In ercises 5 10, use the diagram. (ee ample 1.) 5. Name the circle. K 6. Name two radii. 7. Name two chords. J 8. Name a diameter. H 9. Name a secant. 10. Name a tangent and a point of tangency. 17. 18. In ercises 19, tell whether is tangent to. plain your reasoning. (ee ample 3.) 19. 0. 15 5 9 18 3 4 In ercises 11 14, copy the diagram. ell how many common tangents the circles have and draw them. (ee ample.) 11. 1. 1. 48 60 0. 1 16 8 13. 14. In ercises 15 18, tell whether the common tangent is internal or eternal. 15. 16. In ercises 3 6, point is a point of tangency. ind the radius r of. (ee ample 4.) 3. 5. 14 r r r 7 r 4 16 4. 6. r 6 9 r r 30 r 18 538 hapter 10 ircles
ONUION In ercises 7 and 8, construct with the given radius and point outside of. hen construct a line tangent to that passes through. 7. r = in. 8. r = 4.5 cm In ercises 9 3, points and are points of tangency. ind the value(s) of. (ee ample 5.) 9. 31. + 7 5 8 + 4 30. 3 + 10 3. 7 6 + 5 3 + 7 33. O NYI escribe and correct the error in determining whether XY is tangent to Z. Z 60 Y 11 61 X ecause 11 + 60 = 61, XYZ is a right triangle. o, XY is tangent to Z. 34. O NYI escribe and correct the error in finding the radius of. U 39 36 39 36 = 15 o, the radius is 15. 35. ONIN or a point outside of a circle, how many lines eist tangent to the circle that pass through the point? How many such lines eist for a point on the circle? inside the circle? plain your reasoning. 36. II HINKIN When will two lines tangent to the same circle not intersect? Justify your answer. V 37. UIN UU ach side of quadrilateral VWX is tangent to Y. ind the perimeter of the quadrilateral. X 8.3 1. Y W 4.5 3.1 V 3.3 38. OI In, radii and are perpendicular. and are tangent to. a. ketch,,,, and. b. What type of quadrilateral is? plain your reasoning. 39. KIN N UN wo bike paths are tangent to an approimately circular pond. Your class is building a nature trail that begins at the intersection of the bike paths and runs between the bike paths and over a bridge through the center of the pond. Your classmate uses the onverse of the ngle isector heorem (heorem 6.4) to conclude that the trail must bisect the angle formed by the bike paths. Is your classmate correct? plain your reasoning. 40. OIN WIH HI bicycle chain is pulled tightly so that N is a common tangent of the gears. ind the distance between the centers of the gears. 1.8 in. 17.6 in. 4.3 in. 41. WIIN plain why the diameter of a circle is the longest chord of the circle. N ection 10.1 ines and egments hat Intersect ircles 539
4. HOW O YOU I? In the figure, is tangent to the dime, is tangent to the quarter, and is a common internal tangent. How do you know that? 43. OO In the diagram, is a common internal tangent to and. rove that =. 46. OVIN HO rove the ternal angent ongruence heorem (heorem 10.). iven and are tangent to. rove 47. OVIN HO Use the diagram to prove each part of the biconditional in the angent ine to ircle heorem (heorem 10.1). 44. HOUH OVOKIN polygon is circumscribed about a circle when every side of the polygon is tangent to the circle. In the diagram, quadrilateral is circumscribed about. Is it always true that + = +? Justify your answer. Z 45. HI ONNION ind the values of and y. Justify your answer. 5 + 8 Y W X 4y 1 + 6 m a. rove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. (Hint: If you assume line m is not perpendicular to, then the perpendicular segment from point to line m must intersect line m at some other point.) iven ine m is tangent to at point. rove m b. rove indirectly that if a line is perpendicular to a radius at its endpoint, then the line is tangent to the circle. iven m rove ine m is tangent to. 48. ONIN In the diagram, = = 1, = 8, and all three segments are tangent to. What is the radius of? Justify your answer. aintaining athematical roficiency ind the indicated measure. (ection 1. and ection 1.5) 49. m JK K 15 J 8 50. eviewing what you learned in previous grades and lessons 10 7 540 hapter 10 ircles
10. X NI KNOW N KI.3..1. inding rc easures ssential uestion How are circular arcs measured? central angle of a circle is an angle whose verte is the center of the circle. circular arc is a portion of a circle, as shown below. he measure of a circular arc is the measure of its central angle. If m O < 180, then the circular arc is called a minor arc and is denoted by. circular arc 59 O central angle m = 59 easuring ircular rcs Work with a partner. Use dynamic geometry software to find the measure of. Verify your answers using trigonometry. a. 6 4 oints (0, 0) (5, 0) (4, 3) b. 6 4 oints (0, 0) (5, 0) (3, 4) 0 6 4 0 4 6 0 6 4 0 4 6 4 4 6 6 c. 6 4 oints (0, 0) (4, 3) (3, 4) d. 6 4 oints (0, 0) (4, 3) ( 4, 3) 0 6 4 0 4 6 0 6 4 0 4 6 IN OO o be proficient in math, you need to use technological tools to eplore and deepen your understanding of concepts. 4 6 ommunicate Your nswer. How are circular arcs measured? 3. Use dynamic geometry software to draw a circular arc with the given measure. 4 6 a. 30 b. 45 c. 60 d. 90 ection 10. inding rc easures 541
10. esson What You Will earn ore Vocabulary central angle, p. 54 minor arc, p. 54 major arc, p. 54 semicircle, p. 54 measure of a minor arc, p. 54 measure of a major arc, p. 54 adjacent arcs, p. 543 congruent circles, p. 544 congruent arcs, p. 544 similar arcs, p. 545 ind arc measures. Identify congruent arcs. rove circles are similar. inding rc easures 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 180, then the points on that lie in the interior of form a minor arc with endpoints and. he points on that do not lie on the minor arc form a major arc with endpoints and. semicircle is an arc with endpoints that are the endpoints of a diameter. major arc minor arc UY I he measure of a minor arc is less than 180. he measure of a major arc is greater than 180. inor arcs are named by their endpoints. he minor arc associated with is named. ajor arcs and semicircles are named by their endpoints and a point on the arc. he major arc associated with can be named. ore oncept easuring rcs he measure of a minor arc is the measure of its central angle. he epression m is read as the measure of arc. he measure of the entire circle is 360. he measure of a major arc is the difference of 360 and the measure of the related minor arc. he measure of a semicircle is 180. 50 m = 50 m = 360 50 = 310 inding easures of rcs ind the measure of each arc of, where is a diameter. a. b. c. 110 OUION a. is a minor arc, so m = m = 110. b. is a major arc, so m = 360 110 = 50. c. is a diameter, so is a semicircle, and m = 180. 54 hapter 10 ircles
wo arcs of the same circle are adjacent arcs when they intersect at eactly one point. You can add the measures of two adjacent arcs. ostulate ostulate 10.1 rc ddition ostulate he measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m + m Using the rc ddition ostulate ind the measure of each arc. a. b. c. OUION a. m = m H + m H = 40 + 80 = 10 b. m = m + m = 10 + 110 = 30 c. m = 360 m = 360 30 = 130 inding easures of rcs H 40 80 110 recent survey asked teenagers whether they would rather meet a famous musician, athlete, actor, inventor, or other person. he circle graph shows the results. ind the indicated arc measures. a. m c. m OUION a. m = m + m b. m d. m b. m = m + m = 9 + 108 = 137 + 83 = 137 = 0 c. m = 360 m d. m = 360 m = 360 137 = 360 61 = 3 = 99 onitoring rogress Identify the given arc as a major arc, minor arc, or semicircle. hen find the measure of the arc. 1. 4.. 5. Help in nglish and panish at igideasath.com 3. 6. Whom Would You ather eet? Inventor: 98 usician: 1088 ctor: 798 thlete: 838 Other: 618 10 80 60 ection 10. inding rc easures 543
Identifying ongruent rcs wo circles are congruent circles if and only if a rigid motion or a composition of rigid motions maps one circle onto the other. his statement is equivalent to the ongruent ircles heorem below. heorem heorem 10.3 ongruent ircles heorem wo circles are congruent circles if and only if they have the same radius. roof. 35, p. 548 if and only if. wo arcs are congruent arcs if and only if they have the same measure and they are arcs of the same circle or of congruent circles. heorem heorem 10.4 ongruent entral ngles heorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. roof. 37, p. 548 if and only if. Identifying ongruent rcs ell whether the red arcs are congruent. plain why or why not. a. 80 80 b. c. U U 95 V Y 95 X Z UY I he two circles in part (c) are congruent by the ongruent ircles heorem because they have the same radius. OUION a. because they are arcs of the same circle and m = m. lso, by the ongruent entral ngles heorem. b. and U have the same measure, but are not congruent because they are arcs of circles that are not congruent. c. UV YZ because they are arcs of congruent circles and m UV = m YZ. lso, UV YXZ by the ongruent entral ngles heorem. 544 hapter 10 ircles
onitoring rogress Help in nglish and panish at igideasath.com ell whether the red arcs are congruent. plain why or why not. 7. 145 145 8. N 10 10 5 4 roving ircles re imilar heorem heorem 10.5 imilar ircles heorem ll circles are similar. roof p. 545;. 33, p. 548 imilar ircles heorem ll circles are similar. iven with center and radius r, with center and radius s s r rove irst, translate so that point maps to point. he image of is with center. o, and are concentric circles. r s s r circle is the set of all points that are r units from point. ilate using center of dilation and scale factor s r. s r circle s his dilation maps the set of all the points that are r units from point to the set of all points that are s (r) = s units from point. is the set of all points that are s units r from point. o, this dilation maps to. ecause a similarity transformation maps to,. wo arcs are similar arcs if and only if they have the same measure. ll congruent arcs are similar, but not all similar arcs are congruent. or instance, in ample 4, the pairs of arcs in parts (a), (b), and (c) are similar but only the pairs of arcs in parts (a) and (c) are congruent. ection 10. inding rc easures 545
10. ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. VOUY opy and complete: If and are congruent central angles of, then and are.. WHIH ON ON ON? Which circle does not belong with the other three? plain your reasoning. 6 in. 1 ft 1 in. 6 in. onitoring rogress and odeling with athematics In ercises 3 6, name the red minor arc and find its measure. hen name the blue major arc and find its measure. 3. 5. J 135 10 4. K 6. 170 68 In ercises 7 14, identify the given arc as a major arc, minor arc, or semicircle. hen find the measure of the arc. (ee ample 1.) 7. 8. 9. 11. 1. 13. 14. 10. 1108 708 708 458 658 N In ercises 15 and 16, find the measure of each arc. (ee ample.) 15. a. J b. K c. J d. J 16. a. b. c. d. J K 538 688 798 48 17. OIN WIH HI recent survey asked high school students their favorite type of music. he results are shown in the circle graph. ind each indicated arc measure. (ee ample 3.) avorite ype of usic ountry: &: 178 op: H 668 558 Hip-Hop/ap: 68 88 898 olk: Other: 38 478 hristian: ock: a. m b. m c. m d. m H e. m f. m 546 hapter 10 ircles
18. ONIN he circle graph shows the percentages of students enrolled in fall sports at a high school. Is it possible to find the measure of each minor arc? If so, find the measure of the arc for each category shown. If not, eplain why it is not possible. High chool all ports V ootball: 0% None: 15% W ross-ountry: 0% Z Y Volleyball: 15% occer: 30% In ercises 19, tell whether the red arcs are congruent. plain why or why not. (ee ample 4.) 19. 1808 708 408 1. V. 98 8 W 0. X Y 98 16 10 1 1808 1808 X Z H 858 HI ONNION In ercises 3 and 4, find the value of. hen find the measure of the red arc. 3. 8 ( 30)8 4. 48 78 68 N 78 6. KIN N UN Your friend claims that there is not enough information given to find the value of. Is your friend correct? plain your reasoning. 8 48 N 8 7. O NYI escribe and correct the error in naming the red arc. 708 8. O NYI escribe and correct the error in naming congruent arcs. J N O K JK N 9. NIN O IION wo diameters of are and. ind m and m when m = 0. 30. ONIN In, m = 60, m = 5, m = 70, and m = 0. ind two possible measures of. 31. OIN WIH HI On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle? 5. KIN N UN Your friend claims that any two arcs with the same measure are similar. Your cousin claims that any two arcs with the same measure are congruent. Who is correct? plain. ection 10. inding rc easures 547
3. OIN WIH HI You can use the time zone wheel to find the time in different locations across the world. or eample, to find the time in okyo when it is 4 p.m. in an rancisco, rotate the small wheel until 4 p.m. and an rancisco line up, as shown. hen look at okyo to see that it is 9 a.m. there... 5 4 6 7 9 8 anila angkok stana 3 okyo 10 ashkent Yakutsk oscow 11 ydney ome Kuwait ity Helsinki Noon 1 1 idnight 1 nadyr elfast Wellington zores 1 Honolulu nchorage an rancisco ernando de Noronha 11 10 enver 3 Halifa 9 4 New Orleans ayenne oston 5 8.. 6 7 34. ONIN Is there enough information to tell whether? plain your reasoning. 35. OVIN HO Use the diagram on page 544 to prove each part of the biconditional in the ongruent ircles heorem (heorem 10.3). a. iven b. iven rove rove 36. HOW O YOU I? re the circles on the target similar or congruent? plain your reasoning. a. What is the arc measure between each time zone on the wheel? b. What is the measure of the minor arc from the okyo zone to the nchorage zone? c. If two locations differ by 180 on the wheel, then it is 3 p.m. at one location when it is at the other location. 33. OVIN HO Write a coordinate proof of the imilar ircles heorem (heorem 10.5). iven O with center O(0, 0) and radius r, with center (a, 0) and radius s rove O O y r s 37. OVIN HO Use the diagram to prove each part of the biconditional in the ongruent entral ngles heorem (heorem 10.4). a. iven rove b. iven rove 38. HOUH OVOKIN Write a formula for the length of a circular arc. Justify your answer. aintaining athematical roficiency eviewing what you learned in previous grades and lessons ind the value of. ell whether the side lengths form a ythagorean triple. (ection 9.1) 39. 40. 41. 4. 17 13 8 13 7 11 14 10 548 hapter 10 ircles
NYZIN HI IONHI 10.3 X NI KNOW N KI.5..1. o be proficient in math, you need to look closely to discern a pattern or structure. Using hords ssential uestion What are two ways to determine when a chord is a diameter of a circle? rawing iameters Work with a partner. Use dynamic geometry software to construct a circle of radius 5 with center at the origin. raw a diameter that has the given point as an endpoint. plain how you know that the chord you drew is a diameter. a. (4, 3) b. (0, 5) c. ( 3, 4) d. ( 5, 0) Work with a partner. Use dynamic geometry software to construct a chord of a circle. onstruct a chord on the perpendicular bisector of. What do you notice? hange the original chord and the circle several times. re your results always the same? Use your results to write a conjecture. Writing a onjecture about hords hord erpendicular to a iameter Work with a partner. Use dynamic geometry software to construct a diameter of a circle. hen construct a chord perpendicular to at point. ind the lengths and. What do you notice? hange the chord perpendicular to and the circle several times. o you always get the same results? Write a conjecture about a chord that is perpendicular to a diameter of a circle. ommunicate Your nswer 4. What are two ways to determine when a chord is a diameter of a circle? ection 10.3 Using hords 549
10.3 esson What You Will earn ore Vocabulary revious chord arc diameter IN If, then the point, and any line, segment, or ray that contains, bisects. Use chords of circles to find lengths and arc measures. Using hords of ircles 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. heorems heorem 10.6 ongruent orresponding hords heorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. roof. 19, p. 554 semicircle diameter semicircle major arc chord minor arc if and only if. bisects. heorem 10.7 erpendicular hord isector heorem If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. H roof., p. 554 If is a diameter and, then H H and. heorem 10.8 erpendicular hord isector onverse If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. roof. 3, p. 554 If is a perpendicular bisector of, then is a diameter of the circle. Using ongruent hords to ind an rc easure In the diagram,, JK, J and m JK = 80. ind m. 80 K OUION ecause and JK are congruent chords in congruent circles, the corresponding minor arcs and JK are congruent by the ongruent orresponding hords heorem. o, m = m JK = 80. 550 hapter 10 ircles
Using a iameter a. ind HK. b. ind m HK. 11 J (70 + ) K N 7 OUION H a. iameter J is perpendicular to HK. o, by the erpendicular hord isector hoerem, J bisects HK, and HN = NK. o, HK = (NK) = (7) = 14. b. iameter J is perpendicular to HK. o, by the erpendicular hord isector heorem, J bisects HK, and m HJ = m JK. m HJ = m JK erpendicular hord isector heorem 11 = (70 + ) ubstitute. 10 = 70 ubtract from each side. = 7 ivide each side by 10. o, m HJ = m JK = (70 + ) = (70 + 7) = 77, and m HK = (m HJ ) = (77 ) = 154. Using erpendicular isectors hree bushes are arranged in a garden, as shown. Where should you place a sprinkler so that it is the same distance from each bush? OUION tep 1 tep tep 3 sprinkler abel the bushes,, and, as shown. raw segments and. raw the perpendicular bisectors of and. y the erpendicular hord isector onverse, these lie on diameters of the circle containing,, and. ind the point where the perpendicular bisectors intersect. his is the center of the circle, which is equidistant from points,, and. onitoring rogress In ercises 1 and, use the diagram of. 1. If m = 110, find m.. If m = 150, find m. In ercises 3 and 4, find the indicated length or arc measure. 3. 4. m Help in nglish and panish at igideasath.com 5 9 (80 ) 9 9 ection 10.3 Using hords 551
heorem heorem 10.9 quidistant hords heorem In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. roof. 5, p. 554 if and only if =. Using ongruent hords to ind a ircle s adius W U 5 9 Y V X In the diagram, = = 16, U =, and V = 5 9. ind the radius of. OUION ecause is a segment whose endpoints are the center and a point on the circle, it is a radius of. ecause U, U is a right triangle. pply properties of chords to find the lengths of the legs of U. W U radius 5 9 Y V X tep 1 ind U. ecause and are congruent chords, and are equidistant from by the quidistant hords heorem. o, U = V. U = V quidistant hords heorem = 5 9 ubstitute. = 3 olve for. o, U = = (3) = 6. tep ind U. ecause diameter WX, WX bisects by the erpendicular hord isector heorem. o, U = 1 (16) = 8. J 3 K N 7 1 tep 3 ind. ecause the lengths of the legs are U = 6 and U = 8, U is a right triangle with the ythagorean triple 6, 8, 10. o, = 10. o, the radius of is 10 units. onitoring rogress Help in nglish and panish at igideasath.com 5. In the diagram, JK = = 4, N = 3, and N = 7 1. ind the radius of N. 55 hapter 10 ircles
10.3 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. WIIN escribe what it means to bisect a chord.. WIIN wo chords of a circle are perpendicular and congruent. oes one of them have to be a diameter? plain your reasoning. onitoring rogress and odeling with athematics In ercises 3 6, find the measure of the red arc or chord in. (ee ample 1.) 3. 5. 110 Z 60 75 W Y X 10 4. 34 5 U 34 V 6. 7 N 7 10 11 In ercises 7 10, find the value of. (ee ample.) 7. 9. J 8 H N + 9 8. U 40 5 6 10. H (5 + ) (7 1) 11. O NYI escribe and correct the error in reasoning. ecause bisects,. 1. O OVIN In the cross section of the submarine shown, the control panels are parallel and the same length. escribe a method you can use to find the center of the cross section. Justify your method. (ee ample 3.) In ercises 13 and 14, determine whether is a diameter of the circle. plain your reasoning. 13. 14. 3 3 In ercises 15 and 16, find the radius of. (ee ample 4.) 15. 16 16. 4 + 4 4 + 3 7 6 H 16 ection 10.3 Using hords 553 5 5 5 6 6 17. O OVIN n archaeologist finds part of a circular plate. What was the 7 in. diameter of the plate to the nearest tenth of an 7 in. inch? Justify your answer. 6 in. 6 in.
18. HOW O YOU I? What can you conclude from each diagram? Name a theorem that justifies your answer. a. c. H J b. d. N 90 90 19. OVIN HO Use the diagram to prove each part of the biconditional in the ongruent orresponding hords heorem (heorem 10.6). a. iven and are congruent chords. rove b. iven rove 0. HI ONNION In, all the arcs shown have integer measures. how that must be even.. OVIN HO Use congruent triangles to prove the erpendicular hord isector heorem (heorem 10.7). iven is a diameter of. rove, 3. OVIN HO Write a proof of the erpendicular hord isector onverse (heorem 10.8). iven is a perpendicular bisector of. rove is a diameter of the circle. (Hint: lot the center and draw and.) 4. HOUH OVOKIN onsider two chords that intersect at point. o you think that =? Justify your answer. 5. OVIN HO Use the diagram with the quidistant hords heorem (heorem 10.9) on page 55 to prove both parts of the biconditional of this theorem. 6. KIN N UN car is designed so that the rear wheel is only partially visible below the body of the car. he bottom edge of the panel is parallel to the ground. Your friend claims that the point where the tire touches the ground bisects. Is your friend correct? plain your reasoning. 1. ONIN In, the lengths of the parallel chords are 0, 16, and 1. ind m. plain your reasoning. aintaining athematical roficiency ind the missing interior angle measure. (ection 7.1) eviewing what you learned in previous grades and lessons 7. uadrilateral JK has angle measures m J = 3, m K = 5, and m = 44. ind m. 8. entagon has angle measures m = 85, m = 134, m = 97, and m = 10. ind m. 554 hapter 10 ircles
10.1 10.3 What id You earn? ore Vocabulary circle, p. 534 center, p. 534 radius, p. 534 chord, p. 534 diameter, p. 534 secant, p. 534 tangent, p. 534 point of tangency, p. 534 tangent circles, p. 535 concentric circles, p. 535 common tangent, p. 535 central angle, p. 54 minor arc, p. 54 major arc, p. 54 semicircle, p. 54 measure of a minor arc, p. 54 measure of a major arc, p. 54 adjacent arcs, p. 543 congruent circles, p. 544 congruent arcs, p. 544 similar arcs, p. 545 ore oncepts ection 10.1 ines and egments hat Intersect ircles, p. 534 oplanar ircles and ommon angents, p. 535 heorem 10.1 angent ine to ircle heorem, p. 536 ection 10. easuring rcs, p. 54 ostulate 10.1 rc ddition ostulate, p. 543 heorem 10.3 ongruent ircles heorem, p. 544 ection 10.3 heorem 10.6 ongruent orresponding hords heorem, p. 550 heorem 10.7 erpendicular hord isector heorem, p. 550 heorem 10. ternal angent ongruence heorem, p. 536 heorem 10.4 ongruent entral ngles heorem, p. 544 heorem 10.5 imilar ircles heorem, p. 545 heorem 10.8 erpendicular hord isector onverse, p. 550 heorem 10.9 quidistant hords heorem, p. 55 athematical hinking 1. plain how separating quadrilateral VWX into several segments helped you solve ercise 37 on page 539.. In ercise 30 on page 547, what two cases did you consider to reach your answers? re there any other cases? plain your reasoning. 3. plain how you used inductive reasoning to solve ercise 4 on page 554. tudy kills Keeping Your ind ocused While ompleting Homework efore doing homework, review the oncept boes and amples. alk through the amples out loud. omplete homework as though you were also preparing for a quiz. emorize the different types of problems, formulas, rules, and so on. 555
10.1 10.3 uiz In ercises 1 6, use the diagram. (ection 10.1) 1. Name the circle.. Name a radius. 3. Name a diameter. 4. Name a chord. J N 5. Name a secant. 6. Name a tangent. ind the value of. (ection 10.1) K 7. 9 15 8. 6 3 3 + 18 Identify the given arc as a major arc, minor arc, or semicircle. hen find the measure of the arc. (ection 10.) 9. 10. 11. 13. 1. 14. ell whether the red arcs are congruent. plain why or why not. (ection 10.) 67 70 36 15. J K 16. 7 98 98 15 17. ind the measure of the red arc in. (ection 10.3) + 5 3 1 H J 18. In the diagram, = = 30, = + 5, and J = 3 1. ind the radius of. (ection 10.3) 1 110 150 H 1 19. circular clock can be divided into 1 congruent sections. (ection 10.) a. ind the measure of each arc in this circle. b. ind the measure of the minor arc formed by the hour and minute hands when the time is 7:00. c. ind a time at which the hour and minute hands form an arc that is congruent to the arc in part (b). 556 hapter 10 ircles
10.4 X NI KNOW N KI.5..1. Inscribed ngles and olygons ssential uestion How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. n arc that lies between two lines, rays, or segments is called an intercepted arc. polygon is an inscribed polygon when all its vertices lie on a circle. central angle intercepted arc inscribed angle O Inscribed ngles and entral ngles Work with a partner. Use dynamic geometry software. UIN I HI NU o be proficient in math, you need to communicate precisely with others. a. onstruct an inscribed angle in a circle. hen construct the corresponding central angle. b. easure both angles. How is the inscribed angle related to its intercepted arc? c. epeat parts (a) and (b) several times. ecord your results in a table. Write a conjecture about how an inscribed angle is related to its intercepted arc. ample uadrilateral with Inscribed ngles Work with a partner. Use dynamic geometry software. a. onstruct a quadrilateral with each verte on a circle. ample b. easure all four angles. What relationships do you notice? c. epeat parts (a) and (b) several times. ecord your results in a table. hen write a conjecture that summarizes the data. ommunicate Your nswer 3. How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? 4. uadrilateral H is inscribed in, and m = 80. What is m? plain. ection 10.4 Inscribed ngles and olygons 557
10.4 esson What You Will earn ore Vocabulary inscribed angle, p. 558 intercepted arc, p. 558 subtend, p. 558 inscribed polygon, p. 560 circumscribed circle, p. 560 Use inscribed angles. Use inscribed polygons. Using Inscribed ngles ore oncept Inscribed ngle and Intercepted rc n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. n arc that lies between two lines, rays, or segments is called an intercepted arc. If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle. inscribed angle intercepts. subtends. subtends. intercepted arc heorem heorem 10.10 easure of an Inscribed ngle heorem he measure of an inscribed angle is one-half the measure of its intercepted arc. roof. 37, p. 564 m = 1 m he proof of the easure of an Inscribed ngle heorem involves three cases. ase 1 enter is on a side of the inscribed angle. ase enter is inside the inscribed angle. ase 3 enter is outside the inscribed angle. ind the indicated measure. a. m b. m Using Inscribed ngles 48 OUION a. m = 1 m = 1 (48 ) = 4 b. m = m = 50 = 100 ecause is a semicircle, m = 180 m = 180 100 = 80. 50 558 hapter 10 ircles
inding the easure of an Intercepted rc ind m and m. What do you notice about and U? OUION rom the easure of an Inscribed ngle heorem, you know that m = m U = (31 ) = 6. lso, m = 1 m = 1 (6 ) = 31. 31 U o, U. ample suggests the Inscribed ngles of a ircle heorem. heorem heorem 10.11 Inscribed ngles of a ircle heorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent. roof. 38, p. 564 inding the easure of an ngle iven m = 75, find m. 75 OUION oth and intercept H. o, by the Inscribed ngles of a ircle heorem. o, m = m = 75. onitoring rogress ind the measure of the red arc or angle. H Help in nglish and panish at igideasath.com 1. H 90. V 38 U 3. Y 7 X W Z ection 10.4 Inscribed ngles and olygons 559
Using Inscribed olygons ore oncept Inscribed olygon polygon is an inscribed polygon when all its vertices lie on a circle. he circle that contains the vertices is a circumscribed circle. inscribed polygon circumscribed circle heorems heorem 10.1 Inscribed ight riangle heorem 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. m = 90 if and only if roof. 39, p. 564 is a diameter of the circle. heorem 10.13 Inscribed uadrilateral heorem quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. roof. 40, p. 564,,, and lie on if and only if m + m = m + m = 180. ind the value of each variable. a. OUION Using Inscribed olygons b. z y 10 80 a. is a diameter. o, is a right angle, and m = 90 by the Inscribed ight riangle heorem. = 90 = 45 he value of is 45. b. is inscribed in a circle, so opposite angles are supplementary by the Inscribed uadrilateral heorem. m + m = 180 m + m = 180 z + 80 = 180 10 + y = 180 z = 100 y = 60 he value of z is 100 and the value of y is 60. 560 hapter 10 ircles
onstructing a quare Inscribed in a ircle iven, construct a square inscribed in a circle. OUION tep 1 tep tep 3 raw a diameter raw any diameter. abel the endpoints and. onstruct a perpendicular bisector onstruct the perpendicular bisector of the diameter. abel the points where it intersects as points and. orm a square onnect points,,, and to form a square. Using a ircumscribed ircle Your camera has a 90 field of vision, and you want to photograph the front of a statue. You stand at a location in which the front of the statue is all that appears in your camera s field of vision, as shown. You want to change your location. Where else can you stand so that the front of the statue is all that appears in your camera s field of vision? OUION rom the Inscribed ight riangle heorem, 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. he statue fits perfectly within your camera s 90 field of vision from any point on the semicircle in front of the statue. onitoring rogress Help in nglish and panish at igideasath.com ind the value of each variable. 4. y K 40 5. y 68 8 6. c 10 (c 6) 8 U V 7. In ample 5, eplain how to find locations where the front and left side of the statue are all that appears in your camera s field of vision. ection 10.4 Inscribed ngles and olygons 561
10.4 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. VOUY If a circle is circumscribed about a polygon, then the polygon is an.. IN WO, UION Which is different? ind both answers. ind m. ind m. 5 5 ind m. ind m. onitoring rogress and odeling with athematics In ercises 3 8, find the indicated measure. (ee amples 1 and.) 3. m 4. m 84 5. m N 6. m N 7. m VU 30 V 160 U 8. m WX W 67 70 10 X Y 75 110 In ercises 9 and 10, name two pairs of congruent angles. 9. 10. W X Y Z In ercises 11 and 1, find the measure of the red arc or angle. (ee ample 3.) 11. H 51 1. 45 40 In ercises 13 16, find the value of each variable. (ee ample 4.) 13. 15. y 95 J 54 4b 130 80 K 3a 110 14. m k 16. X 60 60 34 Y 3 y 17. O NYI escribe and correct the error in finding m. 53 m = 53 Z 56 hapter 10 ircles
18. OIN WIH HI carpenter s square is an -shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use the carpenter s square to draw a diameter on the circular piece of wood? (ee ample 5.) ONIN In ercises 5 30, determine whether a quadrilateral of the given type can always be inscribed inside a circle. plain your reasoning. 5. square 6. rectangle 7. parallelogram 8. kite 9. rhombus 30. isosceles trapezoid HI ONNION In ercises 19 1, find the values of and y. hen find the measures of the interior angles of the polygon. 19. 3 6y 1y 0. 4y 9y 14 4 31. OIN WIH HI hree moons,,, and, are in the same circular orbit 100,000 kilometers above the surface of a planet. he planet is 0,000 kilometers in diameter and m = 90. raw a diagram of the situation. How far is moon from moon? 3. OIN WIH HI t the movie theater, you want to choose a seat that has the best viewing angle, so that you can be close to the screen and still see the whole screen without moving your eyes. You previously decided that seat 7 has the best viewing angle, but this time someone else is already sitting there. Where else can you sit so that your seat has the same viewing angle as seat 7? plain. movie screen 1. 6y 4 6y. KIN N UN Your friend claims that. Is your friend correct? plain your reasoning. ow ow ow ow ow ow ow 33. WIIN right triangle is inscribed in a circle, and the radius of the circle is given. plain how to find the length of the hypotenuse. 7 34. HOW O YOU I? et point Y represent your location on the soccer field below. What type of angle is Y if you stand anywhere on the circle ecept at point or point? 3. ONUION onstruct an equilateral triangle inscribed in a circle. 4. ONUION he side length of an inscribed regular heagon is equal to the radius of the circumscribed circle. Use this fact to construct a regular heagon inscribed in a circle. ection 10.4 Inscribed ngles and olygons 563
35. WIIN plain why the diagonals of a rectangle inscribed in a circle are diameters of the circle. 36. HOUH OVOKIN he figure shows a circle that is circumscribed about. Is it possible to circumscribe a circle about any triangle? Justify your answer. 37. OVIN HO If an angle is inscribed in, the center can be on a side of the inscribed angle, inside the inscribed angle, or outside the inscribed angle. rove each case of the easure of an Inscribed ngle heorem (heorem 10.10). a. ase 1 iven is inscribed in. et m =. enter lies on. rove m = 1 m (Hint: how that is isosceles. hen write m in terms of.) b. ase Use the diagram and auiliary line to write iven and rove statements for ase. hen write a proof. c. ase 3 Use the diagram and auiliary line to write iven and rove statements for ase 3. hen write a proof. 38. OVIN HO Write a paragraph proof of the Inscribed ngles of a ircle heorem (heorem 10.11). irst, draw a diagram and write iven and rove statements. 39. OVIN HO he Inscribed ight riangle heorem (heorem 10.1) is written as a conditional statement and its converse. Write a plan for proof for each statement. 40. OVIN HO opy and complete the paragraph proof of the Inscribed uadrilateral heorem (heorem 10.13). iven with inscribed quadrilateral rove m + m = 180, m + m = 180 y the rc ddition ostulate (ostulate 10.1), m + = 360 and m + m = 360. Using the heorem, m = m, m = m, m = m, and m = m. y the ubstitution roperty of quality, m + = 360, so. imilarly,. 41. II HINKIN In the diagram, is a right angle. If you draw the smallest possible circle through tangent to, the circle will intersect at J and at K. ind the eact length of JK. 3 4. II HINKIN You are making a circular cutting board. o begin, you glue eight 1-inch boards together, as shown. hen you draw and cut a circle J with an 8-inch diameter from the boards. K a. H is a diameter of the circular cutting board. Write a proportion relating J and JH. tate a theorem to justify your answer. b. ind J, JH, and J. What is the length of the cutting board seam labeled K? 5 4 H aintaining athematical roficiency olve the equation. heck your solution. (kills eview Handbook) 1 43. 3 = 145 44. = 63 45. 40 = 46. 75 = 1 ( 30) eviewing what you learned in previous grades and lessons 564 hapter 10 ircles
10.5 X NI KNOW N KI.5..1. ngle elationships in ircles ssential uestion When a chord intersects a tangent line or another chord, what relationships eist among the angles and arcs formed? ngles ormed by a hord and angent ine Work with a partner. Use dynamic geometry software. a. onstruct a chord in a circle. t one of the endpoints of the chord, construct a tangent line to the circle. ample b. ind the measures of the two angles formed by the chord and the tangent line. c. ind the measures of the two circular arcs determined by the chord. d. epeat parts (a) (c) several times. ecord your results in a table. hen write a conjecture that summarizes the data. ngles ormed by Intersecting hords Work with a partner. Use dynamic geometry software. KIN HI UN o be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. a. onstruct two chords that intersect inside a circle. b. ind the measure of one of the angles formed by the intersecting chords. c. ind the measures of the arcs intercepted by the angle in part (b) and its vertical angle. What do you observe? d. epeat parts (a) (c) several times. ecord your results in a table. hen write a conjecture that summarizes the data. ample ommunicate Your nswer 1 148 m 3. When a chord intersects a tangent line or another chord, what relationships eist among the angles and arcs formed? 4. ine m is tangent to the circle in the figure at the left. ind the measure of 1. 5. wo chords intersect inside a circle to form a pair of vertical angles with measures of 55. ind the sum of the measures of the arcs intercepted by the two angles. ection 10.5 ngle elationships in ircles 565
10.5 esson What You Will earn ore Vocabulary circumscribed angle, p. 568 revious tangent chord secant ind angle and arc measures. Use circumscribed angles. inding ngle and rc easures heorem heorem 10.14 angent and Intersected hord heorem 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. 1 roof. 33, p. 57 1 m 1 = m m = m 1 inding ngle and rc easures ine m is tangent to the circle. ind the measure of the red angle or arc. a. b. K m 130 1 m J 15 OUION a. m 1 = 1 (130 ) = 65 b. m KJ = (15 ) = 50 onitoring rogress Help in nglish and panish at igideasath.com ine m is tangent to the circle. ind the indicated measure. 1. m 1. m 3. m XY 1 m 10 m 98 Y 80 m X ore oncept Intersecting ines and ircles If two nonparallel lines intersect a circle, there are three places where the lines can intersect. on the circle inside the circle outside the circle 566 hapter 10 ircles
heorems heorem 10.15 ngles Inside the ircle heorem 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. 35, p. 57 heorem 10.16 ngles Outside the ircle heorem 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 1 m 1 = (m + m ), 1 m = (m + m ) 1 1 m 1 = (m m ) roof. 37, p. 57 1 m = (m m ) W 3 Z X Y 1 m 3 = (mxy mwz ) ind the value of. a. 130 J K 156 inding an ngle easure b. 76 178 OUION a. he chords J and K intersect inside the circle. Use the ngles Inside the ircle heorem. = 1 (m J + m K ) = 1 ( 130 + 156 ) = 143 o, the value of is 143. onitoring rogress ind the value of the variable. b. he tangent and the secant intersect outside the circle. Use the ngles Outside the ircle heorem. m = 1 (m m ) = 1 ( 178 76 ) = 51 o, the value of is 51. Help in nglish and panish at igideasath.com 4. y 95 10 5. J K 44 H 30 a ection 10.5 ngle elationships in ircles 567
Using ircumscribed ngles ore oncept ircumscribed ngle circumscribed angle is an angle whose sides are tangent to a circle. circumscribed angle heorem heorem 10.17 ircumscribed ngle heorem he measure of a circumscribed angle is equal to 180 minus the measure of the central angle that intercepts the same arc. roof. 38, p. 57 m = 180 m inding ngle easures ind the value of. a. 135 b. H 30 J OUION a. y definition, m = m = 135. Use the ircumscribed ngle heorem to find m. m = 180 m ircumscribed ngle heorem = 180 135 ubstitute. = 45 o, the value of is 45. ubtract. b. Use the easure of an Inscribed ngle heorem (heorem 10.10) and the ircumscribed ngle heorem to find m J. m J = 1 m easure of an Inscribed ngle heorem m J = 1 m efinition of minor arc m J = 1 (180 m H) ircumscribed ngle heorem m J = 1 (180 30 ) ubstitute. = 1 (180 30) ubstitute. = 75 implify. o, the value of is 75. 568 hapter 10 ircles
odeling with athematics he northern lights are bright flashes of colored light between 50 and 00 miles above arth. flash occurs 150 miles above arth at point. What is the measure of, the portion of arth from which the flash is visible? (arth s radius is approimately 4000 miles.) 4150 mi 4000 mi OON O ecause the value for m is an approimation, use the symbol instead of =. OUION 1. Understand the roblem You are given the approimate radius of arth and the distance above arth that the flash occurs. You need to find the measure of the arc that represents the portion of arth from which the flash is visible.. ake a lan Use properties of tangents, triangle congruence, and angles outside a circle to find the arc measure. 3. olve the roblem ecause and are tangents, and by the angent ine to ircle heorem (heorem 10.1). lso, by the ternal angent ongruence heorem (heorem 10.), and by the efleive roperty of ongruence (heorem.1). o, by the Hypotenuse-eg ongruence heorem (heorem 5.9). ecause corresponding parts of congruent triangles are congruent,. olve right to find that m 74.5. o, m (74.5 ) = 149. m = 180 m m = 180 m 149 180 m 31 m ircumscribed ngle heorem efinition of minor arc ubstitute. olve for m. he measure of the arc from which the flash is visible is about 31. Not drawn to scale 4. ook ack You can use inverse trigonometric ratios to find m and m. m = cos ( 1 4150) 4000 15.5 m = cos ( 1 4150) 4000 15.5 o, m 15.5 + 15.5 = 31, and therefore m 31. onitoring rogress ind the value of. Help in nglish and panish at igideasath.com 400.73 mi Not drawn to scale 4000 mi 6. N K 10 8. You are on top of ount ainier on a clear day. You are about.73 miles above sea level at point. ind m, which represents the part of arth that you can see. 7. 50 ection 10.5 ngle elationships in ircles 569
10.5 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. O H NN oints,,, and are on a circle, and If m = 1 (m m ), then point is the circle.. WIIN plain how to find the measure of a circumscribed angle. onitoring rogress and odeling with athematics intersects at point. In ercises 3 6, line t is tangent to the circle. ind the indicated measure. (ee ample 1.) 3. m 65 t 4. m 5. m 1 6. m 3 t 1 60 t 117 140 In ercises 7 14, find the value of. (ee amples and 3.) 7. 9. 85 114 145 8. K J 30 3 t ( 30) 9 10. U 34 ( + 6) (3 ) W V 11. ( + 30) 1. ( + 70) ( ) Y 15 X (6 11) Z 13. 73 N 14. H 75 17 O NYI In ercises 15 and 16, describe and correct the error in finding the angle measure. 15. 16. 37 U 46 1 70 1 m U = m = 46 o, m U = 46. m 1 = 1 70 = 5 o, m 1 = 5. In ercises 17, find the indicated angle measure. Justify your answer. 10 6 3 60 4 5 10 17. m 1 18. m 19. m 3 0. m 4 1. m 5. m 6 1 570 hapter 10 ircles
3. O OVIN You are flying in a hot air balloon about 1. miles above the ground. ind the measure of the arc that represents the part of arth you can see. he radius of arth is about 4000 miles. (ee ample 4.) Z W X 7. ONIN In the diagram, is tangent to the circle, and KJ is a diameter. What is the range of possible angle measures of J? plain your reasoning. K J 4001. mi Y Not drawn to scale 4000 mi 8. ONIN In the diagram, is any chord that is not a diameter of the circle. ine m is tangent to the circle at point. What is the range of possible values of? plain your reasoning. (he diagram is not drawn to scale.) 4. O OVIN You are watching fireworks over an iego ay as you sail away in a boat. he highest point the fireworks reach is about 0. mile above the bay. Your eyes are about 0.01 mile above the water. t point you can no longer see the fireworks because of the curvature of arth. he radius of arth is about 4000 miles, and is tangent to arth at point. ind m. ound your answer to the nearest tenth. m 9. OO In the diagram, J and N are secant lines that intersect at point. rove that m JN > m JN. J K N Not drawn to scale 5. HI ONNION In the diagram, is tangent to. Write an algebraic epression for m in terms of. hen find m. 7 3 40 6. HI ONNION he circles in the diagram are concentric. Write an algebraic epression for c in terms of a and b. a b c 30. KIN N UN Your friend claims that it is possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? plain your reasoning. 31. ONIN oints and are on a circle, and t is a tangent line containing and another point. a. raw two diagrams that illustrate this situation. b. Write an equation for m in terms of m for each diagram. c. or what measure of can you use either equation to find m? plain. 3. ONIN XYZ is an equilateral triangle inscribed in. is tangent to at point X, is tangent to at point Y, and is tangent to at point Z. raw a diagram that illustrates this situation. hen classify by its angles and sides. Justify your answer. ection 10.5 ngle elationships in ircles 571
33. OVIN HO o prove the angent and Intersected hord heorem (heorem 10.14), you must prove three cases. a. he diagram shows the case where contains the center of the circle. Use the angent ine to ircle heorem (heorem 10.1) to write a paragraph proof for this case. 36. HOUH OVOKIN In the figure, and are tangent to the circle. oint is any point on the major arc formed by the endpoints of the chord. abel all congruent angles in the figure. Justify your reasoning. b. raw a diagram and write a proof for the case where the center of the circle is in the interior of. c. raw a diagram and write a proof for the case where the center of the circle is in the eterior of. 34. HOW O YOU I? In the diagram, television cameras are positioned at and to record what happens on stage. he stage is an arc of. You would like the camera at to have a 30 view of the stage. hould you move the camera closer or farther away? plain your reasoning. 37. OVIN HO Use the diagram below to prove the ngles Outside the ircle heorem (heorem 10.16) for the case of a tangent and a secant. hen copy the diagrams for the other two cases on page 567 and draw appropriate auiliary segments. Use your diagrams to prove each case. 1 38. OVIN HO rove that the ircumscribed ngle heorem (heorem 10.17) follows from the ngles Outside the ircle heorem (heorem 10.16). 80 30 5 In ercises 39 and 40, find the indicated measure(s). Justify your answer. 39. ind m when m WZY = 00. W 40. ind m and m. Z X Y 60 0 J 35. OVIN HO Write a proof of the ngles Inside the ircle heorem (heorem 10.15). iven hords and intersect inside a circle. rove m 1 = 1 (m 1 + m ) 115 H 85 aintaining athematical roficiency olve the equation. (kills eview Handbook) eviewing what you learned in previous grades and lessons 41. + = 1 4. = 1 + 35 43. 3 = + 4 57 hapter 10 ircles
10.6 X NI KNOW N KI.5..1. egment elationships in ircles ssential uestion What relationships eist among the segments formed by two intersecting chords or among segments of two secants that intersect outside a circle? egments ormed by wo Intersecting hords Work with a partner. Use dynamic geometry software. a. onstruct two chords and ample that intersect in the interior of a circle at a point. b. ind the segment lengths,,, and and complete the table. What do you observe? ONIN o be proficient in math, you need to make sense of quantities and their relationships in problem situations. c. epeat parts (a) and (b) several times. Write a conjecture about your results. ecants Intersecting Outside a ircle Work with a partner. Use dynamic geometry software. a. onstruct two secants and ample that intersect at a point outside a circle, as shown. b. ind the segment lengths,,, and, and complete the table. What do you observe? c. epeat parts (a) and (b) several times. Write a conjecture about your results. ommunicate Your nswer 18 9 8 3. What relationships eist among the segments formed by two intersecting chords or among segments of two secants that intersect outside a circle? 4. ind the segment length in the figure at the left. ection 10.6 egment elationships in ircles 573
10.6 esson What You Will earn ore Vocabulary segments of a chord, p. 574 tangent segment, p. 575 secant segment, p. 575 eternal segment, p. 575 Use segments of chords, tangents, and secants. Using egments of hords, angents, and ecants When two chords intersect in the interior of a circle, each chord is divided into two segments that are called segments of the chord. heorem heorem 10.18 egments of hords heorem 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. 19, p. 578 = Using egments of hords ind and JK. + N J K + 4 + 1 OUION NK NJ = N N ( + 4) = ( + 1) ( + ) + 4 = + 3 + 4 = 3 + = ind and JK by substitution. egments of hords heorem ubstitute. implify. ubtract from each side. ubtract 3 from each side. = ( + ) + ( + 1) JK = + ( + 4) = + + + 1 = + + 4 = 7 = 8 o, = 7 and JK = 8. onitoring rogress ind the value of. Help in nglish and panish at igideasath.com 1. 6 4 3. 4 + 1 3 574 hapter 10 ircles
ore oncept angent egment and ecant egment tangent segment is a segment that is tangent to a circle at an endpoint. secant segment is a segment that contains a chord of a circle and has eactly one endpoint outside the circle. he part of a secant segment that is outside the circle is called an eternal segment. heorem heorem 10.19 egments of ecants heorem 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. 0, p. 578 eternal segment tangent segment secant segment is a tangent segment. is a secant segment. is the eternal segment of. = Using egments of ecants ind the value of. OUION 9 10 11 = egments of ecants heorem 9 (11 + 9) = 10 ( + 10) ubstitute. 180 = 10 + 100 implify. 80 = 10 ubtract 100 from each side. 8 = ivide each side by 10. he value of is 8. onitoring rogress Help in nglish and panish at igideasath.com ind the value of. 3. 9 6 5 4. 3 + 1 + 1 ection 10.6 egment elationships in ircles 575
heorem heorem 10.0 egments of ecants and angents heorem 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 s. 1 and, p. 578 = Using egments of ecants and angents ind. 16 NOH WY In ample 3, you can draw segments and. 16 8 OUION = 16 = ( + 8) ubstitute. 56 = + 8 implify. egments of ecants and angents heorem 0 = + 8 56 Write in standard form. = 8 ± 8 4(1)( 56) (1) = 4 ± 4 17 Use uadratic ormula. implify. 8 ecause and intercept the same arc, they are congruent. y the efleive roperty of ongruence (heorem.),. o, by the imilarity heorem (heorem 8.3). You can use this fact to write and solve a proportion to find. Use the positive solution because lengths cannot be negative. o, = 4 + 4 17 1.49, and 1.49. ind the radius of the aquarium tank. OUION = inding the adius of a ircle egments of ecants and angents heorem r r 0 ft 8 ft 0 = 8 (r + 8) ubstitute. 400 = 16r + 64 implify. 336 = 16r ubtract 64 from each side. 1 = r ivide each side by 16. o, the radius of the tank is 1 feet. onitoring rogress ind the value of. 5. 3 1 6. 5 7 Help in nglish and panish at igideasath.com 7. 8. WH I? In ample 4, = 35 feet and = 14 feet. ind the radius of the tank. 10 1 576 hapter 10 ircles
10.6 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. VOUY he part of the secant segment that is outside the circle is called a(n).. WIIN plain the difference between a tangent segment and a secant segment. onitoring rogress and odeling with athematics In ercises 3 6, find the value of. (ee ample 1.) 3. 4. 1 3 10 6 10 18 9 5. 6. 8 15 + 8 1 6 + 3 In ercises 7 10, find the value of. (ee ample.) 7. 8. 5 10 7 6 4 8 9. 10. 45 5 4 + 4 7 50 In ercises 11 14, find the value of. (ee ample 3.) 11. 1. 4 15. O NYI escribe and correct the error in finding. 3 4 5 = 4 = 5 3 4 = 15 = 3.75 16. OIN WIH HI he assini spacecraft is on a mission in orbit around aturn until eptember 017. hree of aturn s moons, ethys, alypso, and elesto, have nearly circular orbits of radius 95,000 kilometers. he diagram shows the positions of the moons and the spacecraft on one of assini s missions. ind the distance from assini to ethys when is tangent to the circular orbit. (ee ample 4.) ethys aturn alypso 83,000 km l elesto assini 03,000 km 7 9 1 13. 1 14. + 4 3 ection 10.6 egment elationships in ircles 577
17. OIN WIH HI he circular stone mound in Ireland called Newgrange has a diameter of 50 feet. passage 6 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. 0. OVIN HO rove the egments of ecants heorem (heorem 10.19). (Hint: raw a diagram and add auiliary line segments to form similar triangles.) 1. OVIN HO Use the angent ine to ircle heorem (heorem 10.1) to prove the egments of ecants and angents heorem (heorem 10.0) for the special case when the secant segment contains the center of the circle. 50 ft. OVIN HO rove the egments of ecants and angents heorem (heorem 10.0). (Hint: raw a diagram and add auiliary line segments to form similar triangles.) 6 ft 3. WIIN UION In the diagram of the water well,,, and are known. Write an equation for using these three measurements. 18. OIN WIH HI You are designing an animated logo for your website. parkles leave point and move to the outer circle along the segments shown so that all of the sparkles reach the outer circle at the same time. parkles travel from point to point at centimeters per second. How fast should sparkles move from point to point N? plain. 19. OVIN HO Write a two-column proof of the egments of hords heorem (heorem 10.18). lan for roof Use the diagram from page 574. raw and. how that and are similar. Use the fact that corresponding side lengths in similar triangles are proportional. aintaining athematical roficiency 4 cm 6 cm 8 cm olve the equation by completing the square. (kills eview Handbook) 4. HOW O YOU I? Which two theorems would you need to use to find? plain your reasoning. 1 5. II HINKIN In the figure, = 1, = 8, = 6, = 4, and is a point of tangency. ind the radius of. 7. + 4 = 45 8. 1 = 8 9. + 1 + 0 = 34 30. 4 + 8 + 44 = 16 N 6. HOUH OVOKIN ircumscribe a triangle about a circle. hen, using the points of tangency, inscribe a triangle in the circle. ust it be true that the two triangles are similar? plain your reasoning. 14 eviewing what you learned in previous grades and lessons 578 hapter 10 ircles
10.7 X NI KNOW N KI.1. ircles in the oordinate lane ssential uestion What is the equation of a circle with center (h, k) and radius r in the coordinate plane? he quation of a ircle with enter at the Origin Work with a partner. Use dynamic geometry software to construct and determine the equations of circles centered at (0, 0) in the coordinate plane, as described below. a. omplete the first two rows of the table for circles with the given radii. omplete the other rows for circles with radii of your choice. b. Write an equation of a circle with center (0, 0) and radius r. adius 1 quation of circle he quation of a ircle with enter (h, k) Work with a partner. Use dynamic geometry software to construct and determine the equations of circles of radius in the coordinate plane, as described below. a. omplete the first two rows of the table for circles with the given centers. omplete the other rows for circles with centers of your choice. b. Write an equation of a circle with center (h, k) and radius. enter (0, 0) (, 0) quation of circle c. Write an equation of a circle with center (h, k) and radius r. eriving the tandard quation of a ircle UIN O-OVIN I o be proficient in math, you need to eplain correspondences between equations and graphs. Work with a partner. onsider a circle with radius r and center (h, k). Write the istance ormula to represent the distance d between a point (, y) on the circle and the center (h, k) of the circle. hen square each side of the istance ormula equation. How does your result compare with the equation you wrote in part (c) of ploration? y (h, k) r (, y) ommunicate Your nswer 4. What is the equation of a circle with center (h, k) and radius r in the coordinate plane? 5. Write an equation of the circle with center (4, 1) and radius 3. ection 10.7 ircles in the oordinate lane 579
10.7 esson What You Will earn ore Vocabulary standard equation of a circle, p. 580 revious completing the square Write and graph equations of circles. Write coordinate proofs involving circles. olve real-life problems using graphs of circles. Writing and raphing quations of ircles et (, y) represent any point on a circle with center at the origin and radius r. y the ythagorean heorem (heorem 9.1), + y = r. his is the equation of a circle with center at the origin and radius r. y r (, y) y ore oncept tandard quation of a ircle et (, y) represent any point on a circle with center (h, k) and radius r. y the ythagorean heorem (heorem 9.1), ( h) + (y k) = r. his is the standard equation of a circle with center (h, k) and radius r. y (, y) r y k (h, k) h Writing the tandard quation of a ircle y Write the standard equation of each circle. a. the circle shown at the left b. a circle with center (0, 9) and radius 4. OUION a. he radius is 3, and the center b. he radius is 4., and the is at the origin. center is at (0, 9). ( h) + (y k) = r tandard equation ( h) + (y k) = r of a circle ( 0) + (y 0) = 3 ubstitute. ( 0) + [y ( 9)] = 4. + y = 9 implify. + (y + 9) = 17.64 he standard equation of he standard equation of the the circle is + y = 9. circle is + (y + 9) = 17.64. onitoring rogress Help in nglish and panish at igideasath.com Write the standard equation of the circle with the given center and radius. 1. center: (0, 0), radius:.5. center: (, 5), radius: 7 580 hapter 10 ircles
Writing the tandard quation of a ircle he point ( 5, 6) is on a circle with center ( 1, 3). Write the standard equation of the circle. ( 5, 6) 6 y 4 OUION ( 1, 3) o write the standard equation, you need to know the values of h, k, and r. o find r, find the distance between the center and the point ( 5, 6) on the circle. 6 4 r = [ 5 ( 1)] + (6 3) istance ormula = ( 4) + 3 implify. = 5 implify. ubstitute the values for the center and the radius into the standard equation of a circle. ( h) + (y k) = r tandard equation of a circle [ ( 1)] + (y 3) = 5 ubstitute (h, k) = ( 1, 3) and r = 5. ( + 1) + (y 3) = 5 implify. he standard equation of the circle is ( + 1) + (y 3) = 5. o complete the square for the epression + b, add the square of half the coefficient of the term b. + b + ( b ) = ( + b ) 4 8 y 8 (4, ) raphing a ircle he equation of a circle is + y 8 + 4y 16 = 0. ind the center and the radius of the circle. hen graph the circle. OUION You can write the equation in standard form by completing the square on the -terms and the y-terms. + y 8 + 4y 16 = 0 quation of circle 8 + y + 4y = 16 8 + 16 + y + 4y + 4 = 16 + 16 + 4 Isolate constant. roup terms. omplete the square twice. ( 4) + (y + ) = 36 actor left side. implify right side. ( 4) + [y ( )] = 6 ewrite the equation to find the center and the radius. he center is (4, ), and the radius is 6. Use a compass to graph the circle. onitoring rogress Help in nglish and panish at igideasath.com 3. he point (3, 4) is on a circle with center (1, 4). Write the standard equation of the circle. 4. he equation of a circle is + y 8 + 6y + 9 = 0. ind the center and the radius of the circle. hen graph the circle. ection 10.7 ircles in the oordinate lane 581
Writing oordinate roofs Involving ircles Writing a oordinate roof Involving a ircle rove or disprove that the point (, ) lies on the circle centered at the origin and containing the point (, 0). OUION he circle centered at the origin and containing the point (, 0) has the following radius. r = ( h) + ( y k) = ( 0) + (0 0) = o, a point lies on the circle if and only if the distance from that point to the origin is. he distance from (, ) to (0, 0) is d = ( 0 ) + ( 0 ) =. o, the point (, ) lies on the circle centered at the origin and containing the point (, 0). onitoring rogress Help in nglish and panish at igideasath.com 5. rove or disprove that the point ( 1, 5 ) lies on the circle centered at the origin and containing the point (0, 1). olving eal-ife roblems Using raphs of ircles he 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. he epicenter is 7 miles away from (,.5). he epicenter is 4 miles away from (4, 6). he epicenter is 5 miles away from (3,.5). OUION 8 4 4 8 y he 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 (,.5) and radius 7 with center (4, 6) and radius 4 with center (3,.5) and radius 5 o find the epicenter, graph the circles on a coordinate plane where each unit corresponds to one mile. ind the point of intersection of the three circles. he epicenter is at about (5, ). onitoring rogress Help in nglish and panish at igideasath.com 6. Why are three seismographs needed to locate an earthquake s epicenter? 58 hapter 10 ircles
10.7 ercises utorial Help in nglish and panish at igideasath.com Vocabulary and ore oncept heck 1. VOUY What is the standard equation of a circle?. WIIN plain why knowing the location of the center and one point on a circle is enough to graph the circle. onitoring rogress and odeling with athematics In ercises 3 8, write the standard equation of the circle. (ee ample 1.) 3. 3 1 3 1 3 y 1 3 5. a circle with center (0, 0) and radius 7 6. a circle with center (4, 1) and radius 5 4. 6 7. a circle with center ( 3, 4) and radius 1 8. a circle with center (3, 5) and radius 7 In ercises 9 11, use the given information to write the standard equation of the circle. (ee ample.) 9. he center is (0, 0), and a point on the circle is (0, 6). 10. he center is (1, ), and a point on the circle is (4, ). 6 y 18. + y + 4 + 1y = 15 In ercises 19, prove or disprove the statement. (ee ample 4.) 19. he point (, 3) lies on the circle centered at the origin with radius 8. 0. he point ( 4, 5 ) lies on the circle centered at the origin with radius 3. 1. he point ( 6, ) lies on the circle centered at the origin and containing the point (3, 1).. he point ( 7, 5 ) lies on the circle centered at the origin and containing the point (5, ). 3. OIN WIH HI city s commuter system has three zones. Zone 1 serves people living within 3 miles of the city s center. Zone serves those between 3 and 7 miles from the center. Zone 3 serves those over 7 miles from the center. (ee ample 5.) Zone 3 11. he center is (0, 0), and a point on the circle is (3, 7). 1. O NYI escribe and correct the error in writing the standard equation of a circle. Zone 1 he standard equation of a circle with center ( 3, 5) and radius 3 is ( 3) + (y 5) = 9. 0 4 mi Zone In ercises 13 18, find the center and radius of the circle. hen graph the circle. (ee ample 3.) 13. + y = 49 14. ( + 5) + (y 3) = 9 15. + y 6 = 7 16. + y + 4y = 3 17. + y 8 y = 16 a. raph this situation on a coordinate plane where each unit corresponds to 1 mile. ocate the city s center at the origin. b. etermine which zone serves people whose homes are represented by the points (3, 4), (6, 5), (1, ), (0, 3), and (1, 6). ection 10.7 ircles in the oordinate lane 583
4. OIN WIH HI elecommunication towers can be used to transmit cellular phone calls. graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). hese towers have a range of about 3 kilometers. a. ketch a graph and locate the towers. re there any locations that may receive calls from more than one tower? plain your reasoning. b. he center of ity is located at (,.5), and the center of ity is located at (5, 4). ach city has a radius of 1.5 kilometers. Which city seems to have better cell phone coverage? plain your reasoning. 5. ONIN ketch the graph of the circle whose equation is + y = 16. hen sketch the graph of the circle after the translation (, y) (, y 4). What is the equation of the image? ake a conjecture about the equation of the image of a circle centered at the origin after a translation m units to the left and n units down. 7. UIN UU he vertices of XYZ are X(4, 5), Y(4, 13), and Z(8, 9). ind the equation of the circle circumscribed about XYZ. Justify your answer. 8. HOUH OVOKIN circle has center (h, k) and contains point (a, b). Write the equation of the line tangent to the circle at point (a, b). HI ONNION In ercises 9 3, use the equations to determine whether the line is a tangent, a secant, a secant that contains the diameter, or none of these. plain your reasoning. 9. ircle: ( 4) + (y 3) = 9 ine: y = 6 30. ircle: ( + ) + (y ) = 16 ine: y = 4 31. ircle: ( 5) + (y + 1) = 4 ine: y = 1 5 3 6. HOW O YOU I? atch each graph with its equation. a. y b. y c. 4 y 4 d. 4 4 y 3. ircle: ( + 3) + (y 6) = 5 4 ine: y = 3 + 33. KIN N UN Your friend claims that the equation of a circle passing through the points ( 1, 0) and (1, 0) is yk + y = 1 with center (0, k). Is your friend correct? plain your reasoning. 34. ONIN our tangent circles are centered on the -ais. he radius of is twice the radius of O. he radius of is three times the radius of O. he 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? plain your reasoning. y. + (y + 3) = 4. ( 3) + y = 4. ( + 3) + y = 4. + (y 3) = 4 O aintaining athematical roficiency Identify the arc as a major arc, minor arc, or semicircle. hen find the measure of the arc. (ection 10.) 35. 37. 39. 36. 38. 40. eviewing what you learned in previous grades and lessons 53 90 5 65 584 hapter 10 ircles
10.4 10.7 What id You earn? ore Vocabulary inscribed angle, p. 558 circumscribed circle, p. 560 secant segment, p. 575 intercepted arc, p. 558 circumscribed angle, p. 568 eternal segment, p. 575 subtend, p. 558 segments of a chord, p. 574 standard equation of a circle, p. 580 inscribed polygon, p. 560 tangent segment, p. 575 ore oncepts ection 10.4 Inscribed ngle and Intercepted rc, p. 558 heorem 10.10 easure of an Inscribed ngle heorem, p. 558 heorem 10.11 Inscribed ngles of a ircle heorem, p. 559 Inscribed olygon, p. 560 heorem 10.1 Inscribed ight riangle heorem, p. 560 heorem 10.13 Inscribed uadrilateral heorem, p. 560 ection 10.5 heorem 10.14 angent and Intersected hord heorem, p. 566 Intersecting ines and ircles, p. 566 heorem 10.15 ngles Inside the ircle heorem, p. 567 ection 10.6 heorem 10.18 egments of hords heorem, p. 574 angent egment and ecant egment, p. 575 heorem 10.19 egments of ecants heorem, p. 575 heorem 10.16 ngles Outside the ircle heorem, p. 567 ircumscribed ngle, p. 568 heorem 10.17 ircumscribed ngle heorem, p. 568 heorem 10.0 egments of ecants and angents heorem, p. 576 ection 10.7 tandard quation of a ircle, p. 580 Writing oordinate roofs Involving ircles, p. 58 athematical hinking 1. What other tools could you use to complete the task in ercise 18 on page 563?. You have a classmate who is confused about why two diagrams are needed in part (a) of ercise 31 on page 571. plain to your classmate why two diagrams are needed. erformance ask ircular otion What do the properties of tangents tell us about the forces acting on a satellite orbiting around arth? How would the path of the satellite change if the force of gravity were removed? o eplore the answers to these questions and more, go to igideasath.com. 585
10 hapter eview 10.1 ines and egments hat Intersect ircles (pp. 533 540) In the diagram, is tangent to at and is tangent to at. ind the value of. = + 5 = 33 ternal angent ongruence heorem (heorem 10.) ubstitute. = 14 olve for. he value of is 14. + 5 33 ell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of. 1. K. N 3. J 4. KN 5. N 6. N J K N ell whether the common tangent is internal or eternal. 7. 8. oints Y and Z are points of tangency. ind the value of the variable. 9. Y 9a 30 10. c Y + 9c + 6 11. W Z 3a X X 9c + 14 Z W X 3 9 Z r r W 1. ell whether is tangent to. plain. 5 48 10 586 hapter 10 ircles
10. inding rc easures (pp. 541 548) ind the measure of each arc of, where N is a diameter. a. N N is a minor arc, so m N = m N = 10. b. N N is a major arc, so m N = 360 10 = 40. c. N N is a diameter, so N is a semicircle, and m N = 180. K 100 10 N Use the diagram above to find the measure of the indicated arc. 13. K 14. 15. K 16. KN ell whether the red arcs are congruent. plain why or why not. 17. Y W X Z 18. 6 95 95 10.3 Using hords (pp. 549 554) In the diagram,,, and m = 75. ind m. ecause and are congruent chords in congruent circles, the corresponding minor arcs and are congruent by the ongruent orresponding hords heorem (heorem 10.6). o, m = m = 75. ind the measure of. 19. 61 0. 65 1. 75 91. In the diagram, N = = 10, JK = 4, and = 6 4. ind the radius of. J 4 N K 6 4 hapter 10 hapter eview 587
10.4 Inscribed ngles and olygons (pp. 557 564) ind the value of each variable. N is inscribed in a circle, so opposite angles are supplementary by the Inscribed uadrilateral heorem (heorem 10.13). m + m N = 180 m + m = 180 3a + 3a = 180 b + 50 = 180 6a = 180 b = 130 a = 30 he value of a is 30, and the value of b is 130. ind the value(s) of the variable(s). 3a 50 b 3a N 3. 40 4. q 100 4r 80 5. K 6. J 14d 70 N 3y 50 4z 7. m 39 V 44 n 8. Y X c 56 Z 10.5 ngle elationships in ircles (pp. 565 57) ind the value of y. he tangent and secant intersect outside the circle, so you can use the ngles Outside the ircle heorem (heorem 10.16). y = 1 (m m ) y = 1 (190 60 ) ubstitute. y = 65 ngles Outside the ircle heorem implify. 190 60 y he value of y is 65. 588 hapter 10 ircles
ind the value of. 9. 30. 31. 50 15 60 40 96 3. ine is tangent to the circle. ind m XYZ. Y X 10 Z 10.6 egment elationships in ircles (pp. 573 578) ind the value of. he chords and H intersect inside the circle, so you can use the egments of hords heorem (heorem 10.18). = H = 3 6 egments of hords heorem ubstitute. 6 3 = 9 implify. H he value of is 9. ind the value of. 33. 34. 6 K 4 J 3 N 6 + 3 35. Z W 1 Y 8 X 36. local park has a circular ice skating rink. You are standing at point, about 1 feet from the edge of the rink. he distance from you to a point of tangency on the rink is about 0 feet. stimate the radius of the rink. r r 0 ft 1 ft hapter 10 hapter eview 589
10.7 ircles in the oordinate lane (pp. 579 584) Write the standard equation of the circle shown. y 6 4 6 4 he radius is 4, and the center is (, 4). ( h) + (y k) = r tandard equation of a circle [ ( ) ] + (y 4) = 4 ubstitute. ( + ) + (y 4) = 16 implify. he standard equation of the circle is ( + ) + (y 4) = 16. Write the standard equation of the circle shown. 37. y 38. 1 y 39. 6 y 4 6 8 4 6 6 4 8 1 6 Write the standard equation of the circle with the given center and radius. 40. center: (0, 0), radius: 9 41. center: ( 5, ), radius: 1.3 4. center: (6, 1), radius: 4 43. center: ( 3, ), radius: 16 44. center: (10, 7), radius: 3.5 45. center: (0, 0), radius: 5. 46. he point ( 7, 1) is on a circle with center ( 7, 6). Write the standard equation of the circle. 47. he equation of a circle is + y 1 + 8y + 48 = 0. ind the center and the radius of the circle. hen graph the circle. 48. rove or disprove that the point (4, 3) lies on the circle centered at the origin and containing the point ( 5, 0). 590 hapter 10 ircles
10 hapter est ind the measure of each numbered angle in. Justify your answer. 1. 1 145. 1 3. 96 36 38 3 1 4. 77 48 1 Use the diagram. 5. =, = 9, and = 3. ind. 6. = 1, = 3, and = 9. ind. 7. = 9 and = 3. ind. 15 8. ketch a pentagon inscribed in a circle. abel the pentagon. escribe the relationship between each pair of angles. plain your reasoning. a. and b. and ind the value of the variable. Justify your answer. 9. 5 4 3 + 6 10. 6 r 1 r 11. rove or disprove that the point (, 1 ) lies on the circle centered at (0, ) and containing the point ( 1, 4). rove the given statement. 1. 13. J 14. J K 15. bank of lighting hangs over a stage. ach light illuminates a circular region on the stage. coordinate plane is used to arrange the lights, using a corner of the stage as the origin. he equation ( 13) + (y 4) = 16 represents the boundary of the region illuminated by one of the lights. hree actors stand at the points (11, 4), (8, 5), and (15, 5). raph the given equation. hen determine which actors are illuminated by the light. 16. 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. o find the radius, accident investigators choose points and on the tire marks. hen the investigators find the midpoint of. Use the diagram to find the radius r of the circle. plain why this method works. b. he formula = 3.87 fr can be used to estimate a car s speed in miles per hour, where f is the coeffi cient of friction and r is the radius of the circle in feet. If f = 0.7, estimate the car s speed in part (a). 80 ft 160 ft 160 ft (r 80) ft r ft r ft hapter 10 hapter est 591 Not drawn to scale