Circle. Relations. 584 Chapter 14 Circle Relationships

Transcription

1 /19/03 1:11 14 age 584 mac27 ac27:dmm_210: ircle elationships > ake this oldable to help you organize information about the material in this chapter. egin with three sheets of plain " by 11" paper. ➊ old in half along the width. ➋ Open and fold the bottom to form a pocket. lue edges. ➌ epeat steps 1 and 2 three times and glue all three pieces together. ➍ abel each pocket with the lesson names. lace an index card in each pocket. ircle elations hips eading and Writing s you read and study the chapter, you can write the main ideas, examples of theorems, and postulates on the index cards. 584 hapter 14 ircle elationships

2 /19/03 1:11 age 585 mac27 ac27:dmm_210: roblem-olving Workshop roject car manufacturer is running a contest to design a logo for its newest model. he rules require that the logo be circular and contain at least one inscribed angle, one tangent to the circle, one secant angle, and one secant-tangent angle. ll angle and segment measurements must be presented with the design. You must also choose a name for the new car along with the logo that meets the design specifications. Working on the roject > Work with one or two other people to create a winning logo. trategies ook for a pattern. hoose a name for the new model. ecide what size you want your logo to be and raw a diagram. draw your circle with the radius you have chosen. etermine how you can include an inscribed angle, a tangent to the circle, a secant angle, and a secant-tangent angle in your design. Use the properties of chords, secants, and tangents to find the measures of all of the angles and segments in your design. Work backward. ake a table. Use an equation. ake a graph. uess and check. echnology ools Use mathematics software to create a design for your logo and to calculate the measurements of all angles and segments. Use drawing software to draw your logo. esearch or more information about logo designs, visit: resenting the roject raw the design for your logo on poster board or place your computergenerated design on poster board. Include the name you have chosen for the new model. lso include the names of the special segments that you used and the measurements of all of the angles and segments. Write a paragraph explaining how you found the measurements. hapter 14 roblem-olving Workshop 585

3 14 1 Inscribed ngles What You ll earn You ll learn to identify and use properties of inscribed angles. Why It s Important rchitecture Inscribed angles are important in the overall symmetry of many ancient structures. ee xercise 8. ecall that a polygon can be inscribed in a circle. n angle can also be inscribed in a circle. n inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. ngle is an inscribed angle. otice that, the vertex of, lies on. he sides of contain chords and. herefore, is an inscribed angle. ach side of the inscribed angle intersects the circle at a point. he two points and form an arc. We say that intercepts, or that is the intercepted arc of. Words: n angle is inscribed if and only if its vertex lies on the circle and its sides contain chords of the circle. efinition of Inscribed ngle odel: ymbols: is inscribed in. xample 1 etermine whether is an inscribed angle. ame the intercepted arc for the angle. entral ngles, esson 11 2 oint, the vertex of, is not on. o, is not an inscribed angle. he intercepted arc of is. Your urn etermine whether each angle is an inscribed angle. ame the intercepted arc for the angle. a. b. O V 586 hapter 14 ircle elationships

4 You can find the measure of an inscribed angle if you know the measure of its intercepted arc. his is stated in the following theorem. heorem 14 1 Words: odel: he degree measure of an inscribed angle equals onehalf the degree measure of its intercepted arc. ymbols: m 1 2 m You can use heorem 14 1 to find the measure of an inscribed angle or the measure of its intercepted arc if one of the measures is known. xamples 2 If m 58, find m. m 1 2 (m ) heorem 14 1 m 1 2 (58) eplace m with 58. m 29 ame ink 3 In the game shown at the right, WZ is equilateral. ind mwz. W eal World mwz 1 2 (mwz ) heorem (mwz ) (mwz ) 120 mwz eplace mwz with 60. ultiply each side by 2. Z hinese heckers Your urn c. If m 80, find m. d. If m 56, find m. esson 14 1 Inscribed ngles 587

5 In, if the measure of O is 74, what is the measure of inscribed angle O? What is the measure of inscribed angle O? otice that both of the inscribed angles intercept the same arc, O. his relationship is stated in heorem O heorem 14 2 Words: If inscribed angles intercept the same arc or congruent arcs, then the angles are congruent. odel: ymbols: xample lgebra ink 4 In, m1 2x and m2 x 14. ind the value of x and 2 both intercept W. W lgebra eview olving quations with the Variable on oth ides, p heorem 14 2 m1 m2 efinition of congruent angles 2x x 14 eplace m1 with 2x and m2 with x 14. 2x x x 14 x ubtract x from each side. x 14 Your urn e. In, m3 3x and m4 2x 9. ind the value of x. 3 4 U uppose is inscribed in and intercepts semicircle Y. ince 1 my 180, m 180 or herefore, is a right angle. his relationship is stated in heorem Y 588 hapter 14 ircle elationships

6 heorem 14 3 Words: If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. odel: ymbols: m 90 xample lgebra ink 5 In, is a diameter. ind the value of x. Inscribed angle intercepts semicircle. y heorem 14 3, is a right angle. herefore, is a right triangle and and are complementary. 1 ( 2x 13) (4x 13) m m x 13 (4x 13) 90 9 x 90 2 efinition of complementary angles ubstitution ombine like terms x ultiply each side by 2 9. x 20 Your urn f. In, is a diameter and m 4x 14. ind the value of x. heck for Understanding ommunicating athematics 1. escribe an intercepted arc of a circle. tate how its measure relates to the measure of an inscribed angle that intercepts it. 2. raw inscribed angle in that has a measure of 100. Include all labels. inscribed angle intercepted arc uided ractice 3. etermine whether W is an inscribed angle. ame the intercepted arc for the angle. (xample 1) W esson 14 1 Inscribed ngles 589

7 ind each measure. (xamples 2 & 3) 4. m 5. m In each circle, find the value of x. (xamples 4 & 5) I 34 (5x 1) 8. rchitecture efer to in the application at the beginning of the lesson. If m 84, find m. (xample 2) (x 20) x xercises ractice omework elp or xercises , 26, 27 2, , 24 ee xamples 4 22, 23 5 xtra ractice ee page 752. etermine whether each angle is an inscribed angle. ame the intercepted arc for the angle Z 11. ind each measure. 12. mi mxw I 13. mi 16. mxv 14. mi mvw 83 Z O V W X In each circle, find the value of x x (x (2x 6) 11) (3x 10) W x (x 4) 590 hapter 14 ircle elationships

8 21. b U Z b x V O 60 (3x 14) I (x 17) 24. In, m1 13x 9 and m2 27x 65. a. ind the value of x. b. ind m1 and m2. c. If m 92, find m. 1 2 pplications and roblem olving eal World 25. iterature Is ante s suggestion in the quote at the right always possible? xplain why or why not. Or draw a triangle inside a semicircle hat would have no right angle. ante, he ivine omedy 26. istory he symbol at the right appears throughout the Visitor enter in exas Washington-on-the-razos tate istorical ark. If, find m. 27. ritical hinking uadrilateral is inscribed in. how that the opposite angles of the quadrilateral are supplementary. Visitor enter, Washington, exas ixed eview tandardized est ractice 28. Use to find cos. ound to four decimal places. (esson 13 5) 29. right cylinder has a base radius of 4 centimeters and a height of 22 centimeters. ind the lateral area of the cylinder to the nearest hundredth. (esson 12 2) 30. ind the area of a 20 sector in a circle with diameter 15 inches. ound to the nearest hundredth. (esson 11 6) 31. rid In tudents are using a slide projector to magnify insects wings. he ratio of actual length to projected length is 1:25. If the projected length of a wing is 8.14 centimeters, what is the actual length in centimeters? ound to the nearest hundredth. (esson 9 1) 32. ultiple hoice olve 2q (lgebra eview) esson 14 1 Inscribed ngles 591

9 14 2 angents to a ircle What You ll earn You ll learn to identify and apply properties of tangents to circles. Why It s Important stronomy cientists use tangents to calculate distances between stars. ee xample 2. tangent is a line that intersects a circle in exactly one point. lso, by definition, a line segment or ray can be tangent to a circle if it is a part of a line that is tangent to the circle. Using tangents, you can find more properties of circles. efinition of a angent Words: odel: In a plane, a line is a tangent if and only if it intersects a circle in exactly one point. ymbols: ine is tangent to. is called the point of tangency. wo special properties of tangency are stated in the theorems below. Words: odel: In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. heorem 14 4 ymbols: If line is tangent to at point, then. he converse of heorem 14 4 is also true. heorem 14 5 Words: In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent. ymbols: If, then is tangent to at point. 592 hapter 14 ircle elationships

10 xample lgebra ink 1 is tangent to at. ind. rom heorem 14 4,. hus, is a right angle, and is a right triangle. () 2 () 2 () 2 ythagorean heorem 12 cm () eplace with 9 and with 12. () quare 9 and 12. () ake the square root of each side cm Your urn a. is tangent to at. ind In the following activity, you ll find a relationship between two tangents that are drawn from a point outside a circle. aper olding aterials: compass patty paper straightedge tep 1 tep 2 Use a compass to draw a circle on patty paper. raw a point outside the circle. tep 3 tep 4 arefully fold the paper so that a tangent is formed from the point to one side of the circle. Use a straightedge to draw the segment. ark your point of tangency. epeat tep 3 for a tangent line that intersects the tangent line in tep 3. ry hese 1. old the paper so that one tangent covers the other. ompare their lengths. 2. ake a conjecture about the relationship between two tangents drawn from a point outside a circle. esson 14 2 angents to a ircle 593

11 he results of the activity suggest the following theorem. heorem 14 6 Words: odel: If two segments from the same exterior point are tangent to a circle, then they are congruent. ymbols: If and are tangent to, then. eal World xample stronomy ink 2 he ring of stars in the photograph appeared after the small blue galaxy on the right crashed through the large galaxy on the left. he two galaxies are 168 thousand light-years apart ( 168 thousand light-years), and has a radius of 75 thousand light-years. ind and if they are tangent to. artwheel alaxy xplore lan rom heorem 14 6,, so we only need to find the measure of one of the segments. y heorem 14 4,. hus, is a right angle and is a right triangle. We can use the ythagorean heorem to find. olve () 2 () 2 () 2 ythagorean heorem () 2 ubstitution 28, () 2 quare 168 and , () ubtract 5625 from 22,599 () 2 each side. 22,599 () 2 ake the square root of each side xamine heck your answer by substituting into the original equation. () 2 () 2 () ,224 28, he answer checks. If you round your final answer to the nearest tenth, the measure of is about thousand light-years. y heorem 14 6, the measure of is also about thousand light-years. 594 hapter 14 ircle elationships

12 Your urn b. and are tangent to. ind the value of x. (2x 13) cm 47 cm wo circles can be tangent to each other. If two circles are tangent and one circle is inside the other, the circles are internally tangent. If two circles are tangent and neither circle is inside the other, the circles are externally tangent. heck for Understanding ommunicating athematics 1. etermine how many tangents can be drawn to a circle from a single point outside the circle. xplain why these tangents must be congruent. 2. xplain why is tangent to, but is not tangent to. tangent point of tangency internally tangent externally tangent uided ractice etting eady ample: valuate each expression. ound to the nearest tenth. olution: is tangent to at. 7. and are tangent to O. ind to the nearest tenth. ind. (xample 2) (xample 1) 10.5 mm 7 in. O 25 in. 8 mm 8. usic he figure at the right shows a compact disc () packaged in a square case. (xample 2) a. Obtain a case and measure to the nearest centimeter from the corner of the disc case to each point of tangency, such as and. b. Which theorem is verified by your measures? esson 14 2 angents to a ircle 595

13 xercises ractice omework elp or xercises ee xamples 9 11, 15, 16, , 17 20, 2 22, 27 xtra ractice ee page 752. ind each measure. If necessary, round to the nearest tenth. ssume segments that appear to be tangent are tangent m 8 cm 17 cm 26 ft ft cm 8 ft 5 ft 5 in. 2 in. 4 in. In the figure, and are both tangent to. ind each measure. 15. m 16. m ind the perimeter of quadrilateral. xplain how you found the missing measures. 4 ft 6 ft 1 8 ft 4 3 ft I and are tangent to. 20. If I 3x 6 and 9, find the value of x. 21. If mi x and mi 2x 3, find the value of x. I upply a reason to support each statement. 22. I 23. I I xercises hapter 14 ircle elationships

14 pplications and roblem olving eal World 26. cience he science experiment at the right demonstrates zero gravity. When the frame is dropped, the pin rises to pop the balloon. If the pin is 2 centimeters long, find x, the distance the pin must rise to pop the balloon. ound to the nearest tenth. pin x 9 cm 30 cm frame 27. lgebra egular pentagon is circumscribed about. his means that each side of the pentagon is tangent to the circle. a. If 12x 30 and 2x 9, find. b. Why is the point of tangency the midpoint of each side? 28. ritical hinking ow many tangents intersect both circles, each at a single point? ake drawings to show your answers. a. b. c. ixed eview 29. In, find muv. (esson 14 1) U V uilding ladder leaning against the side of a house forms a 72 angle with the ground. If the foot of the ladder is 6 feet from the house, find the height that the top of the ladder reaches. ound to the nearest tenth. (esson 13 4) 31. ecreation ow far is the kite off the ground? ound to the nearest tenth. (esson 13 3) 114 m 30? m tandardized est ractice rid In he plans for s. Wathen s new sunroom call for a window in the shape of a regular octagon. What is the measure of one interior angle of the window? (esson 10 2) 33. ultiple hoice In parallelogram V, 4p 9, mv 75, and V 45. What is the value of p? (esson 8 2) esson 14 2 angents to a ircle 597

15 hapter 14 Investigation aterials ruler compass protractor reas of Inscribed and ircumscribed olygons ircles and polygons are paired together everywhere. You can find them in art, advertising, and jewelry designs. ow do you think the area of a circle compares to the area of a regular polygon inscribed in it, or to the area of a regular polygon circumscribed about it? et s find out. Investigate 1. Use construction tools to draw a circle with a radius of 2 centimeters. abel the circle O. 2. ollow these steps to inscribe an equilateral triangle in O. a. raw radius O as shown. ind the area of the circle to the nearest tenth. b. ince there are three sides in a triangle, the measure of a central angle is 360 3, or 120. raw a 120 angle with side O and vertex O. abel point on the circle as shown. O 120 O c. Using O as one side of an angle, draw a second 120 angle as shown at the right. abel point. 120 O d. onnect points,, and. quilateral triangle is inscribed in O. e. Use a ruler to find the measures of one height and base of. hen find and record its area to the nearest tenth. O 598 hapter 14 ircle elationships

16 onstructing erpendicular ine egments, esson ow circumscribe an equilateral triangle about O by constructing a line tangent to O at,, and. 4. ind and record the area of the circumscribed triangle to the nearest tenth. O In this extension, you will compare the areas of regular inscribed and circumscribed polygons to the area of a circle. ake a table like the one below. ecord your triangle information in the first row. egular olygon rea of rea of rea of (rea of Inscribed ircle Inscribed ircumscribed olygon) (rea of (cm 2 ) olygon (cm 2 ) olygon (cm 2 ) ircumscribed olygon) triangle square pentagon hexagon octagon Use a compass to draw four circles congruent to O. ecord their areas in the table. ollow teps 2 and 3 in the Investigation to inscribe and circumscribe each regular polygon listed in the table. ind and record the area of each inscribed and circumscribed polygon. efer to esson 10 5 to review areas of regular polygons. ind the ratios of inscribed polygon area to circumscribed polygon area. ecord the results in the last column of the table. What do you notice? ake a conjecture about the area of inscribed polygons compared to the area of the circle they inscribe. ake a conjecture about the area of circumscribed polygons compared to the area of the circle they circumscribe. resenting Your onclusions ere are some ideas to help you present your conclusions to the class. ake a poster displaying your table and the drawings of your circles and polygons. ummarize your findings about the areas of inscribed and circumscribed polygons. Investigation or more information on inscribed and circumscribed polygons, visit: hapter 14 Investigation he Ins and Outs of olygons 599

17 14 3 ecant ngles What You ll earn You ll learn to find measures of arcs and angles formed by secants. Why It s Important arketing Understanding secant angles can be helpful in locating the source of data on a map. ee xercise 24. circular saw has a flat guide to help cut accurately. he edge of the guide represents a secant segment to the circular blade of the saw. line segment or ray can be a secant of a circle if the line containing the segment or ray is a secant of the circle. heorem 14 7 Words: odel: line or line segment is a secant to a circle if and only if it intersects the circle in two points. ymbols: is a secant of. hord is a secant segment. When two secants intersect, the angles formed are called secant angles. here are three possible cases. ase 1 ase 2 ase 3 Vertex On the ircle Vertex Inside the ircle Vertex Outside the ircle ecant angle intercepts and is an inscribed angle. ecant angle intercepts, and its vertical angle intercepts. ecant angle intercepts and. 600 hapter 14 ircle elationships When a secant angle is inscribed, as in ase 1, recall that its measure is one-half the measure of the intercepted arc. he following theorems state the formulas for ases 2 and 3.

18 heorem 14 8 Words: odel: If a secant angle has its vertex inside a circle, then its degree measure is one-half the sum of the degree measures of the arcs intercepted by the angle and its vertical angle. 1 ymbols: m1 1 2 (m m ) heorem 14 9 Words: odel: If a secant angle has its vertex outside a circle, then its degree measure is one-half the difference of the degree measures of the intercepted arcs. ymbols: m 1 2 (m m ) You can use these theorems to find the measures of arcs and angles formed by secants. xample 1 ind mw. he vertex of W is inside. pply heorem W 12 You also could have used this method to find m. mw 1 2 (mw m) heorem 14 8 mw 1 (12 42) 2 mw 1 (54) or eplace mw with 12 and m with 42. Your urn O a. ind mo esson 14 3 ecant ngles 601

19 eal World xample rt ink 2 xamine the objects in a student s painting at the right. ince they are difficult to identify, the painting is an example of non-objective art. If m 64 and m 19, find m. he vertex of is outside the circle. pply heorem m 1 2 (m m ) heorem (m 19) eplace m with 64 and m with (m 19) ultiply each side by m m dd 19 to each side. 147 m Your urn b. ind m You can also use algebra to solve problems involving secant angles. xample lgebra ink 3 ind m. xplore irst, find the value of x. hen find m. (5x 2) lan he vertex of is inside. pply heorem (3x 2) lgebra eview olving ulti-tep quations, p. 723 olve m 1 2 (m m ) heorem (5x 2 3x 2) ubstitution (8x) 2 implify inside the parentheses. 76 4x implify. 6 4x ivide each side by x hapter 14 ircle elationships

20 he value of x is 19. ow substitute to find m. m 3x 2 3(19) 2 or 55 eplace x with 19. xamine ind m and substitute into the original equation m 1 2 (m m ). he solution checks. Your urn c. ind the value of x. hen find m. (x 38) (x 12) 30 heck for Understanding ommunicating athematics ath ournal 1. etermine the missing information needed for if you want to use heorem 14 9 to find m. 2. xplain how to find m using only the given information xercises 1 2 secant segment secant angles 3. he word secant comes from the atin word secare. Use a dictionary to find the meaning of the word and explain why secant is used for a line that intersects a circle in exactly two points. uided ractice ind each measure. (xamples 1 & 2) 4. m m In each circle, find the value of x. hen find the given measure. (xample 3) 6. m 7. mo 42 2 x (2x 5) O 30 esson 14 3 ecant ngles 603

21 8. ood cook uses secant segments to cut a round pizza into rectangular pieces. If and m 140, find m. (xample 1) Z xercises ractice ind each measure. 9. mz 10. m1 11. m omework elp or xercises 21, , 12, 14, 15, 19, 1, 3 20, 23, 24, 26 10, 11, 13, 16 18, 25 ee xamples 2 xtra ractice ee page Z mi 13. m 80 I 162 O 27 V 10 1 Y 70 W m In each circle, find the value of x. hen find the given measure. 15. mv 16. m 17. mi 76 W 18. m W (x V 16) x 4x 30 2x I (31 x ) 19. m 20. m Z 41 U (4x 6) 43 (3x 4) I 102 O 65 (x 18) (3x 2) 21. If m4 38 and m 38, find m. 22. If m 198 and m 64, find m In a circle, chords and meet at. If m 115, m 6x 16, and m 3x 12. ind x, m, and m. 4 3 xercises hapter 14 ircle elationships

22 pplications and roblem olving eal World arketing he figure at the right is a one-mile circle of an iego used for research and marketing purposes. What is m? 25. istory he gold figurine at the left was made by the ermanic people in the 8th century. ind m ritical hinking In,. ind m m. xercise 25 ixed eview 27. and are tangent to. ind the value of x. (esson 14 2) 76 km (3x 10) km tandardized est ractice 28. pyramid has a height of 12 millimeters and a base with area of 34 square millimeters. What is its volume? (esson 12 5) 29. ind the circumference of a circle whose diameter is 26 meters. ound to the nearest tenth. (esson 11 5) 30. hort esponse ind the area of a trapezoid whose height measures 8 centimeters and whose bases are 11 centimeters and 9 centimeters long. (esson 10 4) 31. ultiple hoice ind the value for y that verifies that the figure is a parallelogram. (esson 8 3) y y > uiz 1 essons 14 1 through etermine whether is an inscribed angle. ame the intercepted arc. (esson 14 1) ind each measure. ssume segments that appear to be tangent are tangent. (esson 14 2) In each circle, find the value of x. hen find the given measure. (esson 14 3) 4. m 5. m 71 (3x 4) x O (4x 1) esson 14 3 ecant ngles 605

23 14 4 ecant-angent ngles What You ll earn You ll learn to find measures of arcs and angles formed by secants and tangents. Why It s Important rchaeology cientists can learn a lot about an ancient civilization by using secant-tangent angles to find pottery measurements. ee xercise 20. When a secant and a tangent of a circle intersect, a secant-tangent angle is formed. his angle intercepts an arc on the circle. he measure of the arc is related to the measure of the secant-tangent angle. here are two ways that secant-tangent angles are formed, as shown below. ase 1 ase 2 Vertex Outside the ircle Vertex On the ircle ecant-tangent angle intercepts and. ecant-tangent angle intercepts. otice that the vertex of a secant-tangent angle cannot lie inside the circle. his is because the tangent always lies outside the circle, except at the single point of contact. he formulas for the measures of these angles are shown in heorems and heorem Words If a secant-tangent angle has its vertex outside the circle, then its degree measure is one-half the difference of the degree measures of the intercepted arcs. If a secant-tangent angle has its vertex on the circle, then its degree measure is one-half the degree measure of the intercepted arc. odels and ymbols m 1 2 (m m ) m 1 2 (m ) 606 hapter 14 ircle elationships

24 xamples lgebra eview valuating xpressions, p is tangent to at. If m = 200, find m. Vertex of the secant-tangent angle is outside of. pply heorem m 1 2 (m m) heorem m 1 (200 50) 2 ubstitution 50 m 1 (150) or is tangent to at. ind m. 100 Vertex of the secant-tangent angle is on. pply heorem m 1 2 (m ) heorem m 1 (100) or 50 2 ubstitution Your urn is tangent to at and is tangent to at. a. ind m. 76 b. ind m. 156 tangent-tangent angle is formed by two tangents. he vertex of a tangent-tangent angle is always outside the circle. Words: he degree measure of a tangent-tangent angle is one-half the difference of the degree measures of the intercepted arcs. odel: heorem ymbols: m 1 2 (m m ) esson 14 4 ecant-angent ngles 607

25 You can use a I 92 calculator to verify the relationship between a tangent-tangent angle and its intercepted arc stated in heorem I 92 utorial ee pp he calculator screen at the right shows an acute angle,. o verify heorem 14 12, you can measure, find the measures of the intercepted arcs, and then perform the calculation. ry hese 1. Use the calculator to construct and label a figure like the one shown above. hen use the ngle tool on the 6 menu to measure. What measure do you get? 2. ow can you use the ngle tool on to find m 6 and m? Use the calculator to find these measures. What are the results? 3. Use the alculate tool on to find 1 2 (m 6 m). ow does the result compare with m from xercise 1? Is your answer in agreement with heorem 14 12? 4. rag point farther away from the center of the circle. escribe how this affects the arc measures and the measure of. 5. uppose you change to an obtuse angle. o the results from xercises 1 3 change? xplain your answer. You can use heorem to solve problems involving tangenttangent angles. eal World xample rchitecture ink 3 In the 15th century, runelleschi, an Italian architect, used his knowledge of mathematics to create a revolutionary design for the dome of a cathedral in lorence. close-up of one of the windows is shown at the right. ind m. 90 is a tangent-tangent angle. pply heorem he uomo, lorence, Italy In order to find m, first find m. m m 360 he sum of the measures of a minor arc and its major arc is 360. m m hapter 14 ircle elationships

26 m 1 2 (m m) heorem m 1 (270 90) 2 ubstitution m 1 (180) or 90 2 Your urn c. ind m. 110 heck for Understanding ommunicating athematics 1. xplain how to find the measure of a tangenttangent angle. secant-tangent angle tangent-tangent angle 2. ame three secant-tangent angles in. 3. In, and are tangents. aria says that if m increases, m increases. Is she correct? ake some drawings to support your conclusion. uided ractice ind the measure of each angle. ssume segments that appear to be tangent are tangent (xample 1) 5. (xample 2) 6. (xample 3) illiards efer to the application at the beginning of the lesson. If x 31 and y 135, find m1, the angle measure of the cue ball s spin. (xample 1) 1 1 x y esson 14 4 ecant-angent ngles 609

27 xercises ractice ind the measure of each angle. ssume segments that appear to be tangent are tangent omework elp or xercises ee xamples 8, 13, 14, , 11, 15, , 12, 16, 22 3 xtra ractice ee page V W W 101 V In, find the value of x. 18. What is m? (2x 8) (x 6) 43 xercises pplications and roblem olving eal World 19. lgebra I is a secant segment, and is tangent to. ind mi in terms of x. (int: irst find mi in terms of x.) I 2x echanics In the piston and rod diagram at the right, the throw arm moves from position to position. ind m hapter 14 ircle elationships

28 21. rchaeology he most commonly found artifact on an archaeological dig is a pottery shard. any clues about a site and the group of people who lived there can be found by studying these shards. he piece at the right is from a round plate. a. If is a tangent at, and m 60, find m. b. uppose an archaeologist uses a tape measure and finds that the distance along the outside edge of the shard is 8.3 centimeters. What was the circumference of the original plate? xplain how you know. 22. ritical hinking and are tangent to. a. If x represents m, what is m in terms of x? b. ind m. 30 c. ind m m. d. Is the sum of the measures of a tangent-tangent angle and the smaller intercepted arc always equal to the sum in part c? xplain. x ixed eview ind each measure. 23. m3 (esson 14 3) 24. and (esson 13 2) cm 45 tandardized est ractice 25. useums museum of miniatures in os ngeles, alifornia, has 2-inch violins that can actually be played. If the 2-inch model represents a 2-foot violin, what is the scale factor of the model to the actual violin? (int: hange feet to inches.) (esson 12 7) 26. hort esponse he perimeter of is 94 centimeters. If and the scale factor of to is 4, find the 3 perimeter of. (esson 9 7) 27. ultiple hoice ind the solution to the system of equations. (lgebra eview) y 3x 5 5x 3y 43 (2, 11) (11, 2) (2, 11) (11, 2) esson 14 4 ecant-angent ngles 611

29 14 5 egment easures What You ll earn You ll learn to find measures of chords, secants, and tangents. In the circle at the right, chords and intersect at. otice the two pairs of segments that are formed by these intersecting chords. Why It s Important rt he opi Indians used special circle segments in their designs and artwork. ee xercise 20. and are segments of. and are segments of. here exists a special relationship for the measures of the segments formed by intersecting chords. his relationship is stated in the following theorem. Words: If two chords of a circle intersect, then the product of the measures of the segments of one chord equals the product of the measures of the segments of the other chord. heorem odel: ymbols: xample lgebra ink lgebra eview olving One-tep quations, p In, find the value of x. heorem x ubstitution 6x 12 6 x x 2 ivide each side by 6. x Your urn a. In, find UW. 1 x 1 6 W U 15 V 612 hapter 14 ircle elationships

30 and are secant segments of. and are the parts of the segments that lie outside the circle. hey are called external secant segments. efinition of xternal ecant egment Words: odel: segment is an external secant segment if and only if it is the part of a secant segment that is outside a circle. and are external secant segments. special relationship between secant segments and external secant segments is stated in the following theorem. heorem Words: odel: If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment. ymbols: In, a similar relationship exists if one segment is a secant and one is a tangent. is a tangent segment. () 2 his result is formally stated in the following theorem. esson 14 5 egment easures 613

31 heorem Words: odel: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment equals the product of the measures of the secant segment and its external secant segment. ymbols: ( ) 2 xamples 2 ind V and V. V heorem (3 9) 3 V 4 ubstitution 12 3 V (V) 3 6 4( V) 4 4 ivide each side by V 9 V V V egment ddition roperty 4 V 9 ubstitution 4 V ubtract 4 from each side. V 5 lgebra ink 3 ind the value of x to the nearest tenth. x x (U) 2 W heorem x 2 (10 10) 10 ubstitution 10 W U 10 x x ake the square root of each side. x 14.1 Use a calculator. Your urn b. ind the value of x to the nearest tenth. c. ind to the nearest tenth x 614 hapter 14 ircle elationships

32 heck for Understanding ommunicating athematics 1. raw and label a circle that fits the following description. external secant segments as center. ontains secant segments and. ontains external secant segments and. is tangent to the circle at. 2. omplete the steps below to prove heorem efer to shown at the right. a. and heorem b. imilarity ostulate c. efinition of imilar olygons d. 3. eon wrote the equation 4 5 3x to find the value of x in the figure at the right. Yoshica wrote the equation 9 4 (3 x) 3. Who wrote the correct equation? xplain x uided ractice 4. ind the value of x. (xample 1) x V ind each measure. If necessary, round to the nearest tenth. 5. O (xample 2) 6. (xample 3) 4 O W ind to the nearest tenth. (xample 1) 6 in. 7 in. 7 in. esson 14 5 egment easures 615

33 xercises ractice omework elp or xercises 8 10, , 18 1, 2 11, 14, 16, 21 ee xamples 12, 13, 15, 19 3 xtra ractice ee page In each circle, find the value of x. If necessary, round to the nearest tenth W 10. ind each measure. If necessary, round to the nearest tenth x x 2x 16 4 I x 1 x V W W V U 17. If 13, 3.2, and 6, find to the nearest tenth. 18. If 7.5, 7, and 6, find to the nearest tenth. pplications and roblem olving eal World 19. pace he space shuttle iscovery is 145 miles above arth. he diameter of arth is about 8000 miles. ow far is its longest line of sight to arth? igure is not drawn to scale. 20. ative merican rt he traditional sun design appears in many phases of opi art and decoration. ind the length of. x 1.75x 14 cm 8 cm 616 hapter 14 ircle elationships

34 21. ritical hinking ind the radius of : a. using the ythagorean heorem. b. using heorem (int: xtend to the other side of.) c. Which method seems more efficient? xplain ixed eview tandardized est ractice 22. In, find the measure of. (esson 14 4) implify. (esson 13 1) In a circle, the measure of chord is 3, the measure of chord is 3, and m = 35. ind m. (esson 11 3) 25. hort esponse etermine whether the face of the jaguar has line symmetry, rotational symmetry, both, or neither. (esson 10 6) 26. hort esponse ketch and label isosceles trapezoid and its median. (esson 8 5) 104 xercise 22 xercise 25 > uiz 2 essons 14 4 and 14 5 ind the measure of each angle. In each circle, find the value of x. (esson 14 4) (esson 14 5) x x stronomy planisphere is a flattened sphere that shows the whole sky. he smaller circle inside the chart is the area of sky that is visible to the viewer. ind the value of x. (esson 14 5) ( x ) 10 9 (2x 7) esson 14 5 egment easures 617

35 14 6 quations of ircles What You ll earn You ll learn to write equations of circles using the center and the radius. Why It s Important eteorology quations of circles are important in helping meteorologists track storms shown on radar. ee xercise 30. In esson 4 6, you learned that the equation of a straight line is linear. In slope-intercept form, this equation is written as y = mx + b. circle is not a straight line, so its equation is not linear. You can use the istance ormula to find the equation of any circle. ircle has its center at (3, 2). It has a radius of 4 units. et (x, y) represent any point on. hen d, the measure of the distance between and, must be equal to the radius, 4. y 4 units (x, y) (3, 2) O x (x x 2 1 ) 2 (y 2 y 1 ) 2 d (x ) 3 2 (y 2) 2 4 (x ) 3 2 (y 2) (x 3) 2 (y 2) 2 16 istance ormula eplace (x 1, y 1 ) with (3, 2) and (x 2, y 2 ) with (x, y). quare each side of the equation. herefore, the equation of the circle with center at (3, 2) and a radius of 4 units is (x 3) 2 (y 2) his result is generalized in the equation of a circle given below. heorem eneral quation of a ircle Words: odel: he equation of a circle with center at (h, k) and a radius of r units is (x h) 2 (y k) 2 r 2. O y r (h, k) x 618 hapter 14 ircle elationships

36 xample 1 Write an equation of a circle with center (1, 2) and a radius of 2 units. 2 units (1, 2) y O x (x h) 2 (y k) 2 r 2 eneral quation of a ircle [x (1)] 2 (y 2) (h, k) (1, 2), r 2 (x 1) 2 (y 2) 2 4 he equation for the circle is (x 1) 2 (y 2) 2 4. Your urn a. Write an equation of a circle with center at (3, 2) and a diameter of 8 units. You can also use the equation of a circle to find the coordinates of its center and the measure of its radius. eal World xample eography ink 2 he lake in rater ake ark was formed thousands of years ago by the explosive collapse of t. azama. If the park entrance is at (0, 0), then the equation of the circle representing the lake is (x 1) 2 (y 11) 2 9. ind the coordinates of its center and the measure of its diameter. ach unit on the grid represents 2 miles y ntrance rater ake ational ark 97 x ewrite the equation in the form (x h) 2 (y k) 2 r 2. [(x (1)] 2 [(y (11)] ince h 1, k 11, and r 3, the center of the circle is at (1, 11). Its radius is 3 miles, so its diameter is 6 miles. rater ake, Oregon Your urn b. ind the coordinates of the center and the measure of the radius of a circle whose equation is x 2 y esson 14 6 quations of ircles 619

37 heck for Understanding ommunicating athematics 1. raw a circle on a coordinate plane. Use a ruler to find its radius and write its general equation. 2. atch each graph below with one of the equations at the right. (1) (x 1) 2 (y 4) 2 5 (2) (x 1) 2 (y 4) 2 5 (3) (x 1) 2 (y 4) 2 25 (4) (x 1) 2 (y 4) 2 25 a. y b. y O x O x 5 5 c. y d. y 5 5 O x O x 3. xplain how you could find the equation of a line that is tangent to the circle whose equation is (x 4) 2 (y 6) 2 9. ath ournal 4. ow could you find the equation of a circle if you are given the coordinates of the endpoints of a diameter? irst, make a sketch of the problem and then list the information that you need and the steps you could use to find the equation. uided ractice etting eady If r represents the radius and d represents the diameter, find each missing measure. ample: d 1 3, r 2? olution: r or hapter 14 ircle elationships 5. r 2 169, d? 6. d 218, r 2? 7. d 2 5, r2? 8. r 2 1, d? 49 6

38 Write an equation of a circle for each center and radius or diameter measure given. (xample 1) 9. (1, 5), d (3, 4), r 2 ind the coordinates of the center and the measure of the radius for each circle whose equation is given. (xample 2) 11. (x 7) 2 (y 5) (x 6) 2 y otany cientists can tell what years had droughts by studying the rings of bald cypress trees. If the radius of a tree in 1612 was 14.5 inches, write an equation that represents the cross section of the tree. ssume that the center is at (0, 0). (xample 1) xercises ractice omework elp or xercises 14 19, 28, 29, 31, 32 ee xamples , 30 2 xtra ractice ee page 753. Write an equation of a circle for each center and radius or diameter measure given. 14. (2, 11), r (4, 2), d (0, 0), r (6, 0), r (1, 1), d (5, 9), d ind the coordinates of the center and the measure of the radius for each circle whose equation is given. 20. (x 9) 2 (y 10) x 2 (y 5) (x 7) 2 (y 3) x y (x 19) 2 y (x 24) 2 (y 8.1) raph each equation on a coordinate plane. 26. (x 5) 2 (y 2) x 2 (y 3) Write an equation of the circle that has a diameter of 12 units and its center at (4, 7). 29. Write an equation of the circle that has its center at (5, 13) and is tangent to the y-axis. esson 14 6 quations of ircles 621

39 pplications and roblem olving eal World 30. eteorology Often when a hurricane is expected, all people within a certain radius are evacuated. ircles around a radar image can be used to determine a safe radius. If an equation of the circle that represents the evacuated area is given by (x 42) 2 (y 11) , find the coordinates of the center and measure of the radius of the evacuated area. Units are in miles. ata Update or the latest information on the percents of international internet users, visit: echnology lthough nglish is the language used by more than half the Internet users, over 56 million people worldwide use a different language, as shown in the circle graph at the right. If the circle displaying the information has a center (0, 3) and a diameter of 7.4 units, write an equation of the circle. Internet anguages (other than nglish) Italian 4% wedish 4% hinese 6% rench 10% erman 13% Others 17% O ource: uro-arketing ssociates y x panish 24% apanese 22% 32. ritical hinking he graphs of x 4 and y 1 are both tangent to a circle that has its center in the fourth quadrant and a diameter of 14 units. Write an equation of the circle. ixed eview 33. ind to the nearest tenth. 34. oys escribe the basic shape (esson 14 5) of the toy as a geometric solid. (esson 12 1) ind the area of a regular pentagon whose perimeter is 40 inches and whose apothems are each 5.5 inches long. (esson 10 5) tandardized est ractice 36. hort esponse ind the values of x and y. (esson 9 3) x 9 y 37. ultiple hoice ind the length of the diagonal of a rectangle whose length is 12 meters and whose width is 4 meters. (esson 6 6) 48 m 160 m 6.9 m 12.6 m 622 hapter 14 ircle elationships

40 eteorologist o you enjoy watching storms? ave you ever wondered why certain areas of the country have more severe weather conditions such as hurricanes or tornadoes? If so, you may want to consider a career as a meteorologist. In addition to forecasting weather, meteorologists apply their research of arth s atmosphere in areas of agriculture, air and sea transportation, and air-pollution control. 1. uppose your home is located at (0, 0) on a coordinate plane. If the eye of the storm, or the storm s center, is located 25 miles east and 12 miles south of you, what are the coordinates of the storm s center? 2. If the storm has a 7-mile radius, write an equation of the circle representing the storm. 3. raph the equation of the circle in xercise 2. bout eteorologists Working onditions may report from radio or television station studios must be able to work as part of a team those not involved in forecasting work regular hours, usually in offices may observe weather conditions and collect data from aircraft ducation high school math and physical science courses bachelor s degree in meteorology master s or h.. degree is required for research positions. mployment 4 out of 10 meteorologists have federal government jobs. overnment osition eginning eterologist ntry-evel Intern ermanent uty asks erformed collect data, perform computations or analysis learn about the Weather ervice s forecasting equipment and procedures handle more complex forecasting jobs areer ata or the latest information about a career as a meteorologist, visit: hapter 14 ath In the Workplace 623

41 14 tudy uide and ssessment Understanding and Using the Vocabulary fter completing this chapter, you should be able to define each term, property, or phrase and give an example or two of each. external secant segment (p. 613) externally tangent (p. 595) inscribed angle (p. 586) intercepted arc (p. 586) internally tangent (p. 595) point of tangency (p. 592) secant angle (p. 600) secant-tangent angle (p. 606) eview ctivities or more review activities, visit: secant segment (p. 600) tangent (p. 592) tangent-tangent angle (p. 607) hoose the term or terms from the list above that best complete each statement. 1. When two secants intersect, the angles formed are called?. 2. he vertex of a(n)? is on the circle and its sides contain chords of the circle. 3. tangent-tangent angle is formed by two?. 4. tangent intersects a circle in exactly one point called the?. 5. he measure of an inscribed angle equals one-half the measure of its?. 6. (n)? is the part of a secant segment that is outside a circle. 7. (n)? is formed by a vertex outside the circle or by a vertex on the circle. 8.? is a line segment that intersects a circle in exactly two points. 9. he measure of a(n)? is always one-half the difference of the measures of the intercepted arcs. 10. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the?. kills and oncepts Objectives and xamples esson 14 1 Identify and use properties of inscribed angles. m 1 2 m 1 2 m hapter 14 ircle elationships 2 1 eview xercises ind each measure. 11. mxz 12. m Y X Z In each circle, find the value of x (x 2) 3x (6x 5) (5x 15)

42 hapter 14 tudy uide and ssessment Objectives and xamples esson 14 2 Identify and apply properties of tangents to circles. If line is tangent to, then. If, then must be tangent to. If and are tangent to, then. esson 14 3 ind measures of arcs and angles formed by secants. m1 1 2 (mwx myz) W X m 1 2 (mwx my) Z 1 Y ind each measure. 18. m 19. m ind the value of x. hen find m. eview xercises ind each measure. ssume segments that appear to be tangent are tangent in. 6 in. 4 m 17. ind m and. (51x 14) x 6 m 29 esson 14 4 ind measures of arcs and angles formed by secants and tangents. m 1 2 (m m) m 1 2 (m ) ind the measure of each angle. ssume segments that appear to be tangent are tangent m 1 2 (m m) hapter 14 tudy uide and ssessment 625

43 hapter 14 tudy uide and ssessment xtra ractice ee pages Objectives and xamples eview xercises esson 14 5 ind measures of chords, secants, and tangents. VY VW VZ VX Y W V X Z (V) 2 VZ VX In each circle, find the value of x. If necessary, round to the nearest tenth ind each measure. If necessary, round to the nearest tenth x x esson 14 6 Write equations of circles using the center and the radius. Write the equation of a circle with center (3, 1) and a radius of 2 units. (x h) 2 (y k) 2 r 2 eneral equation (x 3) 2 (y 1) (h, k) (3, 1); r 2 he equation is (x 3) 2 (y 1) 2 4. O y x Write the equation of a circle for each center and radius or diameter measure given. 27. (3, 2), r (6, 1), r (5, 5), d 4 ind the coordinates of the center and the measure of the radius for each circle whose equation is given. 30. (x 2) 2 (y 3) (x 9) 2 (y 6) (x 5) 2 (y 7) pplications and roblem olving 33. umber lumber yard receives perfectly round logs of raw lumber for further processing. etermine the diameter of the log at the right. (esson 14 1) 2.5 ft 626 hapter 14 ircle elationships 6 ft 34. lgebra ind x. hen find m. (esson 14 3) 3x (x 47) 110

44 14 est 1. ompare and contrast a tangent to a circle and a secant of a circle. 2. raw a circle with the equation (x 1) 2 (y 1) efine the term external secant segment. X O is inscribed in XYZ, m 130, m 100, and mo 50. ind each measure. 4. myxz 5. m 6. mxzy 7. m 8. moz 9. m Y O Z ind each measure. If necessary, round to the nearest tenth. ssume segments that appear to be tangent are tangent. 10. m mxyz X m 15. WX Y 144 Z W X 12 Y 21 Z ind each value of x. hen find the given measure. 11x 16. m 17. Z W 71 x 5 (9x 2) Z Y Write the equation of a circle for each center and radius or diameter measure given. 18. (6, 1), d (3, 7), r ntiques round stained-glass window is divided into three sections, each a different color. In order to replace the damaged middle section, an artist must determine the exact measurements. ind the measure of hapter 14 est 627

45 14 reparing for tandardized ests ight riangle and rigonometry roblems any geometry problems on standardized tests involve right triangles and the ythagorean heorem. he also includes trigonometry problems. emorize these ratios. sin hy o pposite potenuse, cos hy a djacent potenuse, tan o a pposite djacent tandardized tests often use the reek letter (theta) for the measure of an angle. he right triangle and its multiples, like and , occur frequently on standardized tests. Other ythagorean triples, like and , also occur often. emorize them. tate est xample 32-foot telephone pole is braced with a cable that runs from the top of the pole to a point 7 feet from the base. What is the length of the cable rounded to the nearest tenth? 31.2 ft 32.8 ft 34.3 ft 36.2 ft int If no diagram is given, draw one. xample In the figure at the right, is a right angle, is 3 units long, and is 5 units long. If the θ 5 measure of is, what is the value of cos? olution raw a sketch and label the given information. 32 ft int In trigonometry problems, label the triangle with the words opposite, adjacent, and hypotenuse. You can assume that the pole makes a right angle with the ground. In this right triangle, you know the lengths of the two sides. You need to find the length of the hypotenuse. Use the ythagorean heorem. c 2 a 2 + b 2 c a = 32 and b = 7 c = 1024 and 7 2 = 49 c c 1073 Use a calculator. c 32.8 o the nearest tenth, the hypotenuse is 32.8 feet. he answer is. 7 ft olution adjacent o find cos, you need to know the length of the adjacent side. otice that the hypotenuse is 5 and one side is 3, so this is a right triangle. he adjacent side is 4 units. Use the ratio for cos. adjacent cos hy potenuse 4 5 he answer is. θ 5 hypotenuse opposite hapter 14 ircle elationships