10.6 Investigate Segment Lengths

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com Keystrokes Q U S T I O N What is the relationship between the lengths of segments in a circle?

X L O R raw a circle with two chords =.9 =.9 1. 2.2.72.

ST 2 raw secants raw secants ] and ] and label the intersection point.

ST erform calculations alculate the products p and p.

1 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 of segments in a circle? You can use geometry drawing software to find a relationship between the segments formed by two intersecting chords. X L O R raw a circle with two chords =.9 = ST 1 raw a circle raw a circle and choose four points on the circle. Label them,,, and. ST 2 raw secants raw secants ] and ] and label the intersection point. ST Measure segments Note that } and } are chords. Measure }, }, }, and } in your diagram. ST erform calculations alculate the products p and p. R W O N L U S I O N S Use your observations to complete these eercises 1. What do you notice about the products you found in Step? 2. rag points,,, and, keeping point inside the circle. What do you notice about the new products from Step?. Make a conjecture about the relationship between the four chord segments.. Let } Q and } RS be two chords of a circle that intersect at the point T. If T 9, QT, and RT 1, use your conjecture from ercise to find ST. hapter roperties of ircles

. Find Segment Lengths in ircles efore You found angle and arc measures in circles. Now You will find segment lengths in circles. Why? So you can find distances in astronomy, as in ample.

the chord. THORM For Your Notebook THORM.

segments of the other chord. roof:. 21, p. 9 p p lan for roof To prove Theorem.1, construct two similar triangles. The lengths of the corresponding sides are proportional, so } }.

2 . Find Segment Lengths in ircles efore You found angle and arc measures in circles. Now You will find segment lengths in circles. Why? So you can find distances in astronomy, as in ample. 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. THORM For Your Notebook THORM.1 Segments 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. 9 p p lan for roof To prove Theorem.1, construct two similar triangles. The lengths of the corresponding sides are proportional, so } }. y the ross roducts roperty, p p. X M L 1 Find lengths using Theorem.1 LGR Find ML and JK. M NKp NJ NLp NM Use Theorem.1. p( 1 ) ( 1 1)p( 1 2) Substitute Simplify. K 1 2 N L J 1 2 Subtract 2 from each side. 2 Solve for. Find ML and JK by substitution. ML ( 1 2) 1 ( 1 1) JK 1 ( 1 ) Find Segment Lengths in ircles 9

TNGNTS N SNTS secant segment is a segment that contains a chord of a circle, and has eactly one endpoint outside the circle.

eternal segment secant segment tangent segment THORM For Your Notebook THORM.

secant segment and its eternal segment equals the product of the lengths of the other secant segment and its eternal segment. roof:. 2, p.

3 TNGNTS N SNTS 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 THORM For Your Notebook THORM.1 Segments of Secants 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:. 2, p. 9 p p X M L 2 Standardized Test ractice What is the value of? 2 } 9 T S R RQpR RSpRT Use Theorem.1. p ( 1 ) p ( 1 ) Substitute. 1 9 Simplify. 9 Solve for. c The correct answer is. GUI RTI for amples 1 and 2 Find the value(s) of hapter roperties of ircles

equals the square of the length of the tangent segment. roof:. 2, p. 9 2 p X M L Find lengths using Theorem.

4 THORM For Your Notebook THORM.1 Segments of Secants 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:. 2, p. 9 2 p X M L Find lengths using Theorem.1 NOTHR WY For an alternative method for solving the problem in ample, turn to page 9 for the roblem Solving Workshop. Use the figure at the right to find RS. R RQ 2 RSpRT Use Theorem p( 1 ) Substitute Simplify Write in standard form. 1 S T 2 Ï}} 2 2 (1)(22) Use quadratic formula. 2(1) 2 Ï } 17 Simplify. Use the positive solution, because lengths cannot be negative. c So, 2 1 Ï } 17 <.9, and RS <.9. at classzone.com GUI RTI for ample Find the value of etermine which theorem you would use to find. Then find the value of In the diagram for Theorem.1, what must be true about compared to?. Find Segment Lengths in ircles 91

X M L Solve a real-world problem SIN Tethys, alypso, and Telesto are three of Saturn s moons. ach has a nearly circular orbit 29,000 kilometers in radius.

p 2 Use Theorem.1.,000p < 20,000 2 Substitute. < 9,9 Solve for. c assini is about 9,9 kilometers from Tethys. GUI RTI for ample 11.

WS1 for s., 9, and 21 STNRIZ TST RTI s. 2, 1, 2, and 27 1.

5 X M L Solve a real-world problem SIN Tethys, alypso, and Telesto are three of Saturn s moons. ach has a nearly circular orbit 29,000 kilometers in radius. The assini-huygens spacecraft entered Saturn s orbit in July 200. Telesto is on a point of tangency. Find the distance from assini to Tethys. p 2 Use Theorem.1.,000p < 20,000 2 Substitute. < 9,9 Solve for. c assini is about 9,9 kilometers from Tethys. GUI RTI for ample 11. Why is it appropriate to use the approimation symbol < in the last two steps of the solution to ample?. XRISS SKILL RTI HOMWORK KY WORK-OUT SOLUTIONS on p. WS1 for s., 9, and 21 STNRIZ TST RTI s. 2, 1, 2, and VOULRY opy and complete: The part of the secant segment that is outside the circle is called a(n)?. 2. WRITING plain the difference between a tangent segment and a secant segment. XML 1 FINING SGMNT LNGTHS Find the value of. on p. 9 for s hapter roperties of ircles

6 FINING SGMNT LNGTHS Find the value of. XML 2 on p. 90 for s XML on p. 91 for s RROR NLYSIS escribe and correct the error in finding. p F p F p p p 1.7 F FINING SGMNT LNGTHS Find the value of. Round to the nearest tenth MULTIL HOI Which of the following is a possible value of? FINING LNGTHS Find Q. Round your answers to the nearest tenth. 17. N M 1. S 1 R 19. HLLNG In the figure,,,,, and is a point of tangency. Find the radius of (.. Find Segment Lengths in ircles 9

ROLM SOLVING XML on p. 92 for. 20 20. RHOLOGY The circular stone mound in Ireland called Newgrange has a diameter of 20 feet. passage 2 feet long leads toward the center of the mound.

22. WLLS In the diagram of the water well,,, and are known. Write an equation for using these three measurements. 2. ROOF Use Theorem.1 to prove Theorem.

Sparkles leave point and move to the circle along the segments shown so that all of the sparkles reach the circle at the same time. Sparkles travel from point to point at 2 centimeters per second.

7 ROLM SOLVING XML on p. 92 for RHOLOGY The circular stone mound in Ireland called Newgrange has a diameter of 20 feet. passage 2 feet long leads toward the center of the mound. Find the perpendicular distance from the end of the passage to either side of the mound. 21. ROVING THORM.1 Write a two-column proof of Theorem.1. Use similar triangles as outlined in the lan for roof on page WLLS In the diagram of the water well,,, and are known. Write an equation for using these three measurements. 2. ROOF Use Theorem.1 to prove Theorem.1 for the special case when the secant segment contains the center of the circle. F G 2. SHORT RSONS You are designing an animated logo for your website. Sparkles leave point and move to the circle along the segments shown so that all of the sparkles reach the circle at the same time. Sparkles travel from point to point at 2 centimeters per second. How fast should sparkles move from point to point N? plain. 2. ROVING THORM.1 Use the plan to prove Theorem.1. GIVN c } and } are secant segments. ROV c p p lan for roof raw } and }. Show that n and n are similar. Use the fact that corresponding side lengths in similar triangles are proportional. 2. ROVING THORM.1 Use the plan to prove Theorem.1. GIVN c } is a tangent segment. } is a secant segment. ROV c 2 p lan for roof raw } and }. Use the fact that corresponding side lengths in similar triangles are proportional. 9 WORK-OUT SOLUTIONS on p. WS1 STNRIZ TST RTI

27. XTN RSONS In the diagram, } F is a tangent segment, m, m 20, m F 0,,, and. a. Find m. b. Show that n, nf. c. Let F y and F. Use the results of part (b) to write a proportion involving and y.

8 27. XTN RSONS In the diagram, } F is a tangent segment, m, m 20, m F 0,,, and. a. Find m. b. Show that n, nf. c. Let F y and F. Use the results of part (b) to write a proportion involving and y. Solve 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. F 2. HLLNG Stereographic 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. Find the distance d from the point to its corresponding point 9(, 2) on the plane. quator Not drawn to scale (0, 0) d (, ) MIX RVIW RVIW repare for Lesson.7 in s valuate the epression. (p. 7) 29. Ï }} (2) Ï }} 2 1 (2) 1 ( 2 1) 2 1. Ï }}} [22 2 (2)] 2 1 ( 2 ) 2 2. In right n QR, Q, m Q 0, and m R 0. Find QR and R to the nearest tenth. (p. 7). ] F is tangent to ( at. The radius of ( is and F. Find F. (p. 1) Find the indicated measure. } and } are diameters. (p. 9). m. m. m 7. m. m 9. m 1 F 0 etermine whether } is a diameter of the circle. plain. (p. ) 0. R 7 S XTR RTI for Lesson., p. 91 ONLIN QUIZ at classzone.com 9

eternal secant segment. RO L M Use the figure at the right to find RS.

ecause they both intercept the same arc, RQS > RTQ. y the Refleive roperty of ngle ongruence, QRS > TRQ.

RS } RQ RQ } RT }1 1 } 1 c y the ross roducts roperty, 2 1 2. Use the quadratic formula to find that 2 Ï } 17.

9 LSSON. Using LTRNTIV MTHOS nother Way to Solve ample, page 91 MULTIL RRSNTTIONS You can use similar triangles to find the length of an eternal secant segment. RO L M Use the figure at the right to find RS. R 1 S T M T H O Using Similar Triangles ST 1 raw segments } QS and } QT, and identify the similar triangles. ecause they both intercept the same arc, RQS > RTQ. y the Refleive roperty of ngle ongruence, QRS > TRQ. So, nrsq, nrqt by the Similarity ostulate. ST 2 Use a proportion to solve for RS. RS } RQ RQ } RT }1 1 } 1 c y the ross roducts roperty, Use the quadratic formula to find that 2 Ï } 17. Taking the positive solution, 2 1 Ï } 17 and RS.9. R T I 1. WHT IF? Find RQ in the problem above if the known lengths are RS and ST MULTI-ST ROLM opy the diagram.. HOR Find the value of. 7. SGMNTS OF SNTS Use the Segments of Secants Theorem to write an epression for w in terms of, y, and z. a. raw auiliary segments } and }. Name two similar triangles. b. If 1,, and, find. w z y 9 hapter roperties of ircles

10 tension Use after Lesson. 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. For 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. X M L 1 Find 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. ST 1 ST 2 ST raw point. Locate several points 1 centimeter from. Recognize 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 For Your Notebook How to Find a Locus To find the locus of points that satisfy a given condition, use the following steps. ST 1 raw any figures that are given in the statement of the problem. Locate several points that satisfy the given condition. ST 2 ontinue drawing points until you can recognize the pattern. ST raw the locus and describe it in words. tension: Locus 97

LOI STISFYING TWO OR MOR ONITIONS To find the locus of points that satisfy two or more conditions, first find the locus of points that satisfy each condition alone.

11 LOI STISFYING TWO OR MOR ONITIONS 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. X M L 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? ST 1 ST 2 ST 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. So and form the locus of points that satisfy both conditions. RTI XML 1 on p. 97 for s. 1 XML 2 on p. 9 for s. 9 RWING LOUS 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. oint, the locus of points that are at least 1 inch from. Line j, the locus of points that are no more than 1 inch from j WRITING Write a description of the locus. Include a sketch.. oint lies on linel. What is the locus of points onl and cm from?. oint Q lies on line m. What is the locus of points cm from Q and cm from m? 7. oint R is cm from line k. What is the locus of points that are within cm of R, but further than cm from k?. Linesl and m are parallel. oint is cm from both lines. What is the locus of points betweenl and m and no more than cm from? 9. OG LSH 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. 9 hapter roperties of ircles

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.