Geometry hapter 10 esource Masters
onsumable Workbooks Many of the worksheets contained in the hapter esource Masters booklets are available as consumable workbooks. tudy Guide and Intervention Workbook 0-07-860191-6 kills ractice Workbook 0-07-86019-4 ractice Workbook 0-07-860193- eading to Learn Mathematics Workbook 0-07-861061-3 NW F WKK he answers for hapter 10 of these workbooks can be found in the back of this hapter esource Masters booklet. opyright by he McGraw-Hill ompanies, Inc. ll rights reserved. rinted in the United tates of merica. ermission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe s Geometry. ny other reproduction, for use or sale, is prohibited without prior written permission of the publisher. end all inquiries to: he McGraw-Hill ompanies 8787 rion lace olumbus, H 4340-407 IN: 0-07-860187-8 Geometry hapter 10 esource Masters 1 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
ontents Vocabulary uilder................ vii roof uilder...................... i Lesson 10-1 tudy Guide and Intervention........ 541 54 kills ractice....................... 543 ractice........................... 544 eading to Learn Mathematics.......... 545 nrichment......................... 546 Lesson 10- tudy Guide and Intervention........ 547 548 kills ractice....................... 549 ractice........................... 550 eading to Learn Mathematics.......... 551 nrichment......................... 55 Lesson 10-3 tudy Guide and Intervention........ 553 554 kills ractice....................... 555 ractice........................... 556 eading to Learn Mathematics.......... 557 nrichment......................... 558 Lesson 10-4 tudy Guide and Intervention........ 559 560 kills ractice....................... 561 ractice........................... 56 eading to Learn Mathematics.......... 563 nrichment......................... 564 Lesson 10-5 tudy Guide and Intervention........ 565 566 kills ractice....................... 567 ractice........................... 568 eading to Learn Mathematics.......... 569 nrichment......................... 570 Lesson 10-6 tudy Guide and Intervention........ 571 57 kills ractice....................... 573 ractice........................... 574 eading to Learn Mathematics.......... 575 nrichment......................... 576 Lesson 10-7 tudy Guide and Intervention........ 577 578 kills ractice....................... 579 ractice........................... 580 eading to Learn Mathematics.......... 581 nrichment......................... 58 Lesson 10-8 tudy Guide and Intervention........ 583 584 kills ractice....................... 585 ractice........................... 586 eading to Learn Mathematics.......... 587 nrichment......................... 588 hapter 10 ssessment hapter 10 est, Form 1........... 589 590 hapter 10 est, Form.......... 591 59 hapter 10 est, Form.......... 593 594 hapter 10 est, Form.......... 595 596 hapter 10 est, Form.......... 597 598 hapter 10 est, Form 3........... 599 600 hapter 10 pen-nded ssessment..... 601 hapter 10 Vocabulary est/eview...... 60 hapter 10 Quizzes 1 &.............. 603 hapter 10 Quizzes 3 & 4.............. 604 hapter 10 Mid-hapter est............ 605 hapter 10 umulative eview.......... 606 hapter 10 tandardized est ractice 607 608 Unit 3 est/eview (h. 8 10)....... 609 610 tandardized est ractice tudent ecording heet.............. 1 NW...................... 36 Glencoe/McGraw-Hill iii Glencoe Geometry
eacher s Guide to Using the hapter 10 esource Masters he Fast File hapter esource system allows you to conveniently file the resources you use most often. he hapter 10 esource Masters includes the core materials needed for hapter 10. hese materials include worksheets, etensions, and assessment options. he answers for these pages appear at the back of this booklet. ll of the materials found in this booklet are included for viewing and printing in the Geometry eacherworks -M. Vocabulary uilder ages vii viii include a student study tool that presents up to twenty of the key vocabulary terms from the chapter. tudents are to record definitions and/or eamples for each term. You may suggest that students highlight or star the terms with which they are not familiar. WHN U Give these pages to students before beginning Lesson 10-1. ncourage them to add these pages to their Geometry tudy Notebook. emind them to add definitions and eamples as they complete each lesson. Vocabulary uilder ages i include another student study tool that presents up to fourteen of the key theorems and postulates from the chapter. tudents are to write each theorem or postulate in their own words, including illustrations if they choose to do so. You may suggest that students highlight or star the theorems or postulates with which they are not familiar. WHN U Give these pages to students before beginning Lesson 10-1. ncourage them to add these pages to their Geometry tudy Notebook. emind them to update it as they complete each lesson. tudy Guide and Intervention ach lesson in Geometry addresses two objectives. here is one tudy Guide and Intervention master for each objective. WHN U Use these masters as reteaching activities for students who need additional reinforcement. hese pages can also be used in conjunction with the tudent dition as an instructional tool for students who have been absent. kills ractice here is one master for each lesson. hese provide computational practice at a basic level. WHN U hese masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. ractice here is one master for each lesson. hese problems more closely follow the structure of the ractice and pply section of the tudent dition eercises. hese eercises are of average difficulty. WHN U hese provide additional practice options or may be used as homework for second day teaching of the lesson. eading to Learn Mathematics ne master is included for each lesson. he first section of each master asks questions about the opening paragraph of the lesson in the tudent dition. dditional questions ask students to interpret the contet of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques. WHN U his master can be used as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for LL (nglish Language Learner) students. Glencoe/McGraw-Hill iv Glencoe Geometry
nrichment here is one etension master for each lesson. hese activities may etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. hese are not written eclusively for honors students, but are accessible for use with all levels of students. WHN U hese may be used as etra credit, short-term projects, or as activities for days when class periods are shortened. ssessment ptions he assessment masters in the hapter 10 esources Masters offer a wide range of assessment tools for intermediate and final assessment. he following lists describe each assessment master and its intended use. hapter ssessment H Form 1 contains multiple-choice questions and is intended for use with basic level students. Forms and contain multiple-choice questions aimed at the average level student. hese tests are similar in format to offer comparable testing situations. Forms and are composed of freeresponse questions aimed at the average level student. hese tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form 3 is an advanced level test with free-response questions. Grids without aes are provided for questions assessing graphing skills. ll of the above tests include a freeresponse onus question. he pen-nded ssessment includes performance assessment tasks that are suitable for all students. scoring rubric is included for evaluation guidelines. ample answers are provided for assessment. Vocabulary est, suitable for all students, includes a list of the vocabulary words in the chapter and ten questions assessing students knowledge of those terms. his can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate ssessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Mid-hapter est provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. ontinuing ssessment he umulative eview provides students an opportunity to reinforce and retain skills as they proceed through their study of Geometry. It can also be used as a test. his master includes free-response questions. he tandardized est ractice offers continuing review of geometry concepts in various formats, which may appear on the standardized tests that they may encounter. his practice includes multiplechoice, grid-in, and short-response questions. ubble-in and grid-in answer sections are provided on the master. nswers age 1 is an answer sheet for the tandardized est ractice questions that appear in the tudent dition on pages 588 589. his improves students familiarity with the answer formats they may encounter in test taking. he answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Geometry
10 eading to Learn Mathematics Vocabulary uilder his is an alphabetical list of the key vocabulary terms you will learn in hapter 10. s you study the chapter, complete each term s definition or description. emember to add the page number where you found the term. dd these pages to your Geometry tudy Notebook to review vocabulary at the end of the chapter. center Vocabulary erm Found on age efinition/escription/ample Vocabulary uilder central angle chord circle circumference circumscribed diameter inscribed (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Geometry
10 eading to Learn Mathematics Vocabulary uilder (continued) intercepted Vocabulary erm Found on age efinition/escription/ample major arc minor arc pi () point of tangency radius secants semicircle tangent Glencoe/McGraw-Hill viii Glencoe Geometry
10 Learning to ead Mathematics roof uilder his is a list of key theorems and postulates you will learn in hapter 10. s you study the chapter, write each theorem or postulate in your own words. Include illustrations as appropriate. emember to include the page number where you found the theorem or postulate. dd this page to your Geometry tudy Notebook so you can review the theorems and postulates at the end of the chapter. heorem or ostulate heorem 10.1 Found on age escription/illustration/bbreviation roof uilder heorem 10. heorem 10.3 heorem 10.4 heorem 10.5 heorem 10.6 heorem 10.7 (continued on the net page) Glencoe/McGraw-Hill i Glencoe Geometry
10 Learning to ead Mathematics roof uilder (continued) heorem or ostulate heorem 10.8 Found on age escription/illustration/bbreviation heorem 10.9 heorem 10.11 heorem 10.1 heorem 10.13 heorem 10.14 heorem 10.15 Glencoe/McGraw-Hill Glencoe Geometry
10-1 tudy Guide and Intervention ircles and ircumference arts of ircles circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center. segment or line can intersect a circle in several ways. segment with endpoints that are the center of the circle and a point of the circle is a radius. segment with endpoints that lie on the circle is a chord. chord that contains the circle s center is a diameter. ample a. Name the circle. he name of the circle is. F chord:, radius: F, F, F diameter: Lesson 10-1 b. Name radii of the circle.,,, and are radii. c. Name chords of the circle. and are chords. d. Name a diameter of the circle. is a diameter. ercises 1. Name the circle. X. Name radii of the circle. 3. Name chords of the circle. Y 4. Name diameters of the circle. 5. Find if is 18 millimeters. 6. Find and if Y is 10 inches. 7. Is XY? plain. Glencoe/McGraw-Hill 541 Glencoe Geometry
10-1 tudy Guide and Intervention (continued) ircles and ircumference ircumference he circumference of a circle is the distance around the circle. ircumference For a circumference of units and a diameter of d units or a radius of r units, d or r. ample Find the circumference of the circle to the nearest hundredth. r ircumference formula (13) r 13 81.68 Use a calculator. he circumference is about 81.68 centimeters. 13 cm ercises Find the circumference of a circle with the given radius or diameter. ound to the nearest hundredth. 1. r 8 cm. r 3 ft 3. r 4.1 cm 4. d 10 in. 5. d 1 m 6. d 18 yd 3 he radius, diameter, or circumference of a circle is given. Find the missing measures to the nearest hundredth. 7. r 4 cm 8. d 6 ft d, r, 9. r 1 cm 10. d 15 in. d, r, Find the eact circumference of each circle. 11. 1. 5 cm cm 1 cm cm Glencoe/McGraw-Hill 54 Glencoe Geometry
10-1 kills ractice ircles and ircumference For ercises 15, refer to the circle. 1. Name the circle.. Name a radius. 3. Name a chord. 4. Name a diameter. 5. Name a radius not drawn as part of a diameter. 6. uppose the diameter of the circle is 16 centimeters. Find the radius. Lesson 10-1 7. If 11 inches, find. he diameters of F and G are 5 and 6 units, respectively. Find each measure. 8. F 9. F G he radius, diameter, or circumference of a circle is given. Find the missing measures to the nearest hundredth. 10. r 8 cm 11. r 13 ft d, d, 1. d 9 m 13. 35.7 in. r, d, r Find the eact circumference of each circle. 14. 15. 3 cm 15 ft 8 ft Glencoe/McGraw-Hill 543 Glencoe Geometry
10-1 ractice ircles and ircumference For ercises 15, refer to the circle. 1. Name the circle.. Name a radius. L 3. Name a chord. 4. Name a diameter. W 5. Name a radius not drawn as part of a diameter. 6. uppose the radius of the circle is 3.5 yards. Find the diameter. 7. If 19 meters, find LW. he diameters of L and M are 0 and 13 units, respectively. Find each measure if Q 4. 8. LQ 9. M L Q M he radius, diameter, or circumference of a circle is given. Find the missing measures to the nearest hundredth. 10. r 7.5 mm 11. 7.6 yd d, d, r Find the eact circumference of each circle. 1. 13. 7 cm 4 cm K 40 mi 4 mi UNIL For ercises 14 and 15, use the following information. Herman purchased a sundial to use as the centerpiece for a garden. he diameter of the sundial is 9.5 inches. 14. Find the radius of the sundial. 15. Find the circumference of the sundial to the nearest hundredth. Glencoe/McGraw-Hill 544 Glencoe Geometry
10-1 eading to Learn Mathematics ircles and ircumference re-ctivity How far does a carousel animal travel in one rotation? ead the introduction to Lesson 10-1 at the top of page 5 in your tetbook. How could you measure the approimate distance around the circular carousel using everyday measuring devices? eading the Lesson 1. efer to the figure. a. Name the circle. b. Name four radii of the circle. c. Name a diameter of the circle. d. Name two chords of the circle. Q U Lesson 10-1. Match each description from the first column with the best term from the second column. (ome terms in the second column may be used more than once or not at all.) a. a segment whose endpoints are on a circle b. the set of all points in a plane that are the same distance from a given point c. the distance between the center of a circle and any point on the circle d. a chord that passes through the center of a circle e. a segment whose endpoints are the center and any point on a circle f. a chord made up of two collinear radii g. the distance around a circle 3. Which equations correctly epress a relationship in a circle? i. radius ii. diameter iii. chord iv. circle v. circumference. d r. r. d. d. r d F. r G. r H. d 1 r Helping You emember 4. good way to remember a new geometric term is to relate the word or its parts to geometric terms you already know. Look up the origins of the two parts of the word diameter in your dictionary. plain the meaning of each part and give a term you already know that shares the origin of that part. Glencoe/McGraw-Hill 545 Glencoe Geometry
10-1 nrichment he Four olor roblem Mapmakers have long believed that only four colors are necessary to distinguish among any number of different countries on a plane map. ountries that meet only at a point may have the same color provided they do not have an actual border. he conjecture that four colors are sufficient for every conceivable plane map eventually attracted the attention of mathematicians and became known as the four-color problem. espite etraordinary efforts over many years to solve the problem, no definite answer was obtained until the 1980s. Four colors are indeed sufficient, and the proof was accomplished by making ingenious use of computers. he following problems will help you appreciate some of the compleities of the four-color problem. For these maps, assume that each closed region is a different country. 1. What is the minimum number of colors necessary for each map? a. b. c. d. e.. raw some plane maps on separate sheets. how how each can be colored using four colors. hen determine whether fewer colors would be enough. Glencoe/McGraw-Hill 546 Glencoe Geometry
10- tudy Guide and Intervention ngles and rcs ngles and rcs central angle is an angle whose verte is at the center of a circle and whose sides are radii. central angle separates a circle into two arcs, a major arc and a minor arc. H F GF is a minor arc. HG is a major arc. GF is a central angle. G Here are some properties of central angles and arcs. he sum of the measures of the central angles of mh mf mfg mgh 360 a circle with no interior points in common is 360. he measure of a minor arc equals the measure of its central angle. he measure of a major arc is 360 minus the measure of the minor arc. wo arcs are congruent if and only if their corresponding central angles are congruent. he measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. (rc ddition ostulate) ample mf mf In, m 4 and is a diameter. Find m and m. is a central angle and m 4, so m 4. hus m 360 4 or 318. mgf 360 mf F FG if and only if F FG. mf mfg mg Lesson 10- ercises Find each measure. 1. m. mu 3. mq 4. mq 45 60 U Q If m 44, find each measure. 5. m 6. m 7. m 8. m 9. m 10. m Glencoe/McGraw-Hill 547 Glencoe Geometry
10- tudy Guide and Intervention (continued) ngles and rcs rc Length n arc is part of a circle and its length is a part of the circumference of the circle. ample In, m 135, 8, and is a diameter. Find the length of. m 135, so m 135. Using the formula r, the circumference is (8) or 16.o find the length of, write a proportion to compare each part to its whole. length of degree measure of arc roportion circumference degree measure of circle 1 3 5 ubstitution 16 360 (16 )( 135) 360 6 Multiply each side by 16. implify. he length of is 6 or about 18.85 units. ercises he diameter of is 4 units long. Find the length of each arc for the given angle measure. 1. if m 10. if m 10 3. if m 45 4. if m 45 he diameter of is 15 units long and. Find the length of each arc for the given angle measure. 5. if m 70 M 6. N if m 50 N 7. M 8. M if mm 140 Glencoe/McGraw-Hill 548 Glencoe Geometry
10- kills ractice ngles and rcs LG In, and are diameters. Find each measure. 1. m. m (15 3) (7 5) 4 3. m 4. m 5. m 6. m In, mu 40, U, and U. Find each measure. 7. mq 8. mq 9. m 10. m U Q Lesson 10-11. mu 1. m 13. mq 14. mu he diameter of is 18 units long. Find the length of each arc for the given angle measure. N M 15. LM if mlm 100 16. MN if mmn 80 J K L 17. KL if mkl 60 18. NJK if mnk 10 19. KLM if mkm 160 0. JK if mjk 50 Glencoe/McGraw-Hill 549 Glencoe Geometry
10- ractice ngles and rcs LG In Q, and are diameters. Find each measure. (5 3) 1. mq. mq (6 5) (8 1) Q 3. mq 4. mq 5. mq 6. mq In, mgh 38. Find each measure. 7. mf 8. m H G 9. mfg 10. mhg F 11. mfg 1. mg he radius of Z is 13.5 units long. Find the length of each arc for the given angle measure. 13. Q if mqz 10 14. Q if mqz 60 Z Q 15. Q if mz 150 16. Q if mqz 160 HMWK For ercises 17 and 18, refer to the table, which shows the number of hours students at Leland High chool say they spend on homework each night. 17. If you were to construct a circle graph of the data, how many degrees would be allotted to each category? Homework Less than 1 hour 8% 1 hours 9% 3 hours 58% 3 4 hours 3% ver 4 hours % 18. escribe the arcs associated with each category. Glencoe/McGraw-Hill 550 Glencoe Geometry
10- eading to Learn Mathematics ngles and rcs re-ctivity What kinds of angles do the hands on a clock form? ead the introduction to Lesson 10- at the top of page 59 in your tetbook. What is the measure of the angle formed by the hour hand and the minute hand of the clock at 5:00? What is the measure of the angle formed by the hour hand and the minute hand at 10:30? (Hint: How has each hand moved since 10:00?) eading the Lesson 1. efer to. Indicate whether each statement is true or false. a. is a major arc. b. is a semicircle. c. d. and are adjacent arcs. e. is an acute central angle. f. and are supplementary central angles.. efer to the figure in ercise 1. Give each of the following arc measures. a. m b. m c. m d. m e. m f. m g. m h. m 5 Lesson 10-3. Underline the correct word or number to form a true statement. a. he arc measure of a semicircle is (90/180/360). b. rcs of a circle that have eactly one point in common are (congruent/opposite/adjacent) arcs. c. he measure of a major arc is greater than (0/90/180) and less than (90/180/360). d. uppose a set of central angles of a circle have interiors that do not overlap. If the angles and their interiors contain all points of the circle, then the sum of the measures of the central angles is (90/70/360). e. he measure of an arc formed by two adjacent arcs is the (sum/difference/product) of the measures of the two arcs. f. he measure of a minor arc is greater than (0/90/180) and less than (90/180/360). Helping You emember 4. good way to remember something is to eplain it to someone else. uppose your classmate Luis does not like to work with proportions. What is a way that he can find the length of a minor arc of a circle without solving a proportion? Glencoe/McGraw-Hill 551 Glencoe Geometry
10- nrichment urves of onstant Width circle is called a curve of constant width because no matter how you turn it, the greatest distance across it is always the same. However, the circle is not the only figure with this property. he figure at the right is called a euleau triangle. 1. Use a metric ruler to find the distance from to any point on the opposite side.. Find the distance from Q to the opposite side. 3. What is the distance from to the opposite side? Q he euleau triangle is made of three arcs. In the eample shown, Q has center, Q has center, and has center Q. 4. race the euleau triangle above on a piece of paper and cut it out. Make a square with sides the length you found in ercise 1. how that you can turn the triangle inside the square while keeping its sides in contact with the sides of the square. 5. Make a different curve of constant width by starting with the five points below and following the steps given. tep 1: lace he point of your compass on with opening. Make an arc with endpoints and. tep : Make another arc from to that has center. tep 3: ontinue this process until you have five arcs drawn. ome countries use shapes like this for coins. hey are useful because they can be distinguished by touch, yet they will work in vending machines because of their constant width. 6. Measure the width of the figure you made in ercise 5. raw two parallel lines with the distance between them equal to the width you found. n a piece of paper, trace the five-sided figure and cut it out. how that it will roll between the lines drawn. Glencoe/McGraw-Hill 55 Glencoe Geometry
10-3 tudy Guide and Intervention rcs and hords rcs and hords oints on a circle determine both chords and arcs. everal properties are related to points on a circle. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. If all the vertices of a polygon lie on a circle, the polygon is said to be inscribed in the circle and the circle is circumscribed about the polygon. V V if and only if V. V is inscribed in. is circumscribed about V. ample rapezoid is inscribed in. If and m 50, what is m? hords,, and are congruent, so,, and are congruent. m 50, so m m m 50 50 50 150. hen m 360 150 or 10. ercises ach regular polygon is inscribed in a circle. etermine the measure of each arc that corresponds to a side of the polygon. 1. heagon. pentagon 3. triangle 4. square 5. octagon 6. 36-gon Lesson 10-3 etermine the measure of each arc of the circle circumscribed about the polygon. 7. 4 8. U 9. V U 4 V 7 U Glencoe/McGraw-Hill 553 Glencoe Geometry
10-3 tudy Guide and Intervention (continued) rcs and hords iameters and hords In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. W Z X Y If WZ, then X X and W W. If X Y, then. If, then and are equidistant from point. ample In,, 15, and 4. Find. diameter or radius perpendicular to a chord bisects the chord, so is half of. 1 (4) 1 Use the ythagorean heorem to find in. () () () 1 15 144 5 81 9 ythagorean heorem ubstitution Multiply. ubtract 144 from each side. ake the square root of each side. ercises In, 4 and my 45. Find each measure. 1. Q. 3. Q 4. 5. my 7. mx 8. mx 6. m 9. m X Q Y In G, G GU and. Find each measure. 10. U 11. 1. m 13. 14. G 15. m 5 G 3 U 16. chord of a circle 0 inches long is 4 inches from the center of a circle. Find the length of the radius. Glencoe/McGraw-Hill 554 Glencoe Geometry
10-3 kills ractice rcs and hords In H, m 8, mu 8, 46, and U. Find each measure. M 1. U. K H 3. M 4. mhku K U 5. m 6. m 7. m 8. mu he radius of Y is 34, 60, and m 71. Find each measure. 9. m 10. m Y 11. 1. 13. Y 14. Lesson 10-3 In X, LX MX, XY 58, and VW 84. Find each measure. W Y 15. YZ 16. YM L V X M Z 17. MX 18. MZ 19. LV 0. LX Glencoe/McGraw-Hill 555 Glencoe Geometry
10-3 ractice rcs and hords In, mhq 48, HI JK, and J 7.5. Find each measure. 1. mhi. mqi H K 3. mjk 4. HI Q I J 5. I 6. JK he radius of N is 18, NK 9, and m 10. Find each measure. 7. mg 8. mhn K Y N X H 9. mhn 10. HN G he radius of 3, Q, and Q 56. Find each measure. 11. 14. Q Q 1. 16. 13. MNL he base figure in a mandala design is a nine-pointed star. Find the measure of each arc of the circle circumscribed about the star. Glencoe/McGraw-Hill 556 Glencoe Geometry
10-3 eading to Learn Mathematics rcs and hords re-ctivity How do the grooves in a elgian waffle iron model segments in a circle? ead the introduction to Lesson 10-3 at the top of page 536 in your tetbook. What do you observe about any two of the grooves in the waffle iron shown in the picture in your tetbook? eading the Lesson 1. upply the missing words or phrases to form true statements. a. In a circle, if a radius is to a chord, then it bisects the chord and its. b. In a circle or in circles, two are congruent if and only if their corresponding chords are congruent. c. In a circle or in circles, two chords are congruent if they are from the center. d. polygon is inscribed in a circle if all of its lie on the circle. e. ll of the sides of an inscribed polygon are of the circle.. If has a diameter 40 centimeters long, and F 4 centimeters, find each measure. a. b. G c. d. H e. H f. FG 3. In Q, VW and m 70. Find each measure. a. m b. m c. mvw d. mvu G H F K Q W M U V Lesson 10-3 4. Find the measure of each arc of a circle that is circumscribed about the polygon. a. an equilateral triangle b. a regular pentagon c. a regular heagon d. a regular decagon e. a regular dodecagon f. a regular n-gon Helping You emember 5. ome students have trouble distinguishing between inscribed and circumscribed figures. What is an easy way to remember which is which? Glencoe/McGraw-Hill 557 Glencoe Geometry
10-3 nrichment atterns from hords ome beautiful and interesting patterns result if you draw chords to connect evenly spaced points on a circle. n the circle shown below, 4 points have been marked to divide the circle into 4 equal parts. Numbers from 1 to 48 have been placed beside the points. tudy the diagram to see eactly how this was done. 4 18 43 19 41 17 44 0 45 1 40 16 46 39 15 47 3 38 14 37 13 1 5 6 48 4 1 36 3 7 11 35 10 34 4 8 9 33 5 9 8 3 7 31 6 30 1. Use your ruler and pencil to draw chords to connect numbered points as follows: 1 to, to 4, 3 to 6, 4 to 8, and so on. Keep doubling until you have gone all the way around the circle. What kind of pattern do you get?. opy the original circle, points, and numbers. ry other patterns for connecting points. For eample, you might try tripling the first number to get the number for the second endpoint of each chord. Keep special patterns for a possible class display. Glencoe/McGraw-Hill 558 Glencoe Geometry
10-4 tudy Guide and Intervention Inscribed ngles Inscribed ngles n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. In G, inscribed F intercepts F. G Inscribed ngle heorem If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. F mf 1 mf ample In G above, mf 90. Find mf. F is an inscribed angle so its measure is half of the intercepted arc. mf 1 mf 1 (90) or 45 ercises Use for ercises 1 10. In, V and V. 1. Name the intercepted arc for.. Name an inscribed angle that intercepts V. Q V In, mv 10 and m 76. Find each measure. 3. m 4. mv 5. m 6. mv 7. mq 8. mv Lesson 10-4 9. mv 10. mv Glencoe/McGraw-Hill 559 Glencoe Geometry
10-4 tudy Guide and Intervention (continued) Inscribed ngles ngles of Inscribed olygons n inscribed polygon is one whose sides are chords of a circle and whose vertices are points on the circle. Inscribed polygons have several properties. If an angle of an inscribed polygon intercepts a semicircle, the angle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. If is a semicircle, then m 90. For inscribed quadrilateral, m m 180 and m m 180. ample a. m In above, 3 and 5. Find each measure. intercepts a semicircle. herefore is a right angle and m 90. ercises b. Find the measure of each angle or segment for each figure. is a right triangle, so use the ythagorean heorem to find. () () () () 3 5 () 5 9 () 16 4 1. mx, my. 3. m1, m X G W 10 Y 5 F 55 1 40 Z 1 J H 4. m1, m 5., 6. m1, m K U 3 3 1 9 M N Q 30 Z L 65 30 W 1 V Glencoe/McGraw-Hill 560 Glencoe Geometry
10-4 kills ractice Inscribed ngles In, mkl 80, mlm 100, and mmn 60. Find the measure of each angle. 1. m1. m 3. m3 4. m4 K L 1 6 3 4 5 M N 5. m5 6. m6 LG Find the measure of each numbered angle. 7. m1 5, m 8 8. m1 5, m3 3 10, m4 y 7, m6 3y 11 1 J F 1 6 I U 5 H 3 4 G Quadrilateral U is inscribed in such that mu 0 and m 95. Find each measure. 9. m 10. m U Lesson 10-4 11. mu 1. mu 13. mu 14. m Glencoe/McGraw-Hill 561 Glencoe Geometry
10-4 ractice Inscribed ngles In, mwx 104, mwz 88, and mzwy 6. Find the measure of each angle. 1. m1. m 3. m3 4. m4 Y Z 6 1 5 3 4 W X 5. m5 6. m6 LG Find the measure of each numbered angle. 7. m1 5, m 3 8. m1 4 7, m 11, m3 7y 1, m4 y 10 m3 5y 14, m4 3y 8 G U 4 J H 1 3 I 4 1 3 Quadrilateral FGH is inscribed in N such that mfg 97, mgh 117, and mhg 164. Find each measure. 9. m 10. mf H N F 11. mg 1. mh G 13. ILIY In V, point is randomly located so that it does not coincide with points or. If m 140, what is the probability that m 70? 70 V 140 Glencoe/McGraw-Hill 56 Glencoe Geometry
10-4 eading to Learn Mathematics Inscribed ngles re-ctivity How is a socket like an inscribed polygon? ead the introduction to Lesson 10-4 at the top of page 544 in your tetbook. Why do you think regular heagons are used rather than squares for the hole in a socket? Why do you think regular heagons are used rather than regular polygons with more sides? eading the Lesson 1. Underline the correct word or phrase to form a true statement. a. n angle whose verte is on a circle and whose sides contain chords of the circle is called a(n) (central/inscribed/circumscribed) angle. b. very inscribed angle that intercepts a semicircle is a(n) (acute/right/obtuse) angle. c. he opposite angles of an inscribed quadrilateral are (congruent/complementary/supplementary). d. n inscribed angle that intercepts a major arc is a(n) (acute/right/obtuse) angle. e. wo inscribed angles of a circle that intercept the same arc are (congruent/complementary/supplementary). f. If a triangle is inscribed in a circle and one of the sides of the triangle is a diameter of the circle, the diameter is (the longest side of an acute triangle/a leg of an isosceles triangle/the hypotenuse of a right triangle).. efer to the figure. Find each measure. a. m b. m 68 c. m d. m e. m f. m g. m Helping You emember h. m 59 Lesson 10-4 3. good way to remember a geometric relationship is to visualize it. escribe how you could make a sketch that would help you remember the relationship between the measure of an inscribed angle and the measure of its intercepted arc. Glencoe/McGraw-Hill 563 Glencoe Geometry
10-4 nrichment Formulas for egular olygons uppose a regular polygon of n sides is inscribed in a circle of radius r. he figure shows one of the isosceles triangles formed by joining the endpoints of one side of the polygon to the center of the circle. In the figure, s is the length of each side of the regular polygon, and a is the length of the segment from perpendicular to. r a r s s s Use your knowledge of triangles and trigonometry to solve the following problems. 1. Find a formula for in terms of the number of sides n of the polygon.. Find a formula for s in terms of the number of n and r. Use trigonometry. 3. Find a formula for a in terms of n and r. Use trigonometry. 4. Find a formula for the perimeter of the regular polygon in terms of n and r. Glencoe/McGraw-Hill 564 Glencoe Geometry
10-5 tudy Guide and Intervention angents angents tangent to a circle intersects the circle in eactly one point, called the point of tangency. here are three important relationships involving tangents. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent to the circle. If two segments from the same eterior point are tangent to a circle, then they are congruent. if and only if is tangent to. If and are tangent to, then. ample is tangent to. Find. is tangent to, so is perpendicular to radius. is a radius, so 8 and 9 8 or 17. Use the ythagorean heorem with right. 9 8 () () () 8 17 64 89 5 15 ythagorean heorem ubstitution Multiply. ubtract 64 from each side. ake the square root of each side. ercises Find. ssume that segments that appear to be tangent are tangent. 1.. J 0 K 19 15 H G F 3. N 4. 1 M U Q 30 40 40 5. 6. Y 1 8 Z 5 F 8 Lesson 10-5 Glencoe/McGraw-Hill 565 Glencoe Geometry
10-5 tudy Guide and Intervention (continued) angents ircumscribed olygons When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle. G H F Q Heagon F is circumscribed about.,,,, F, and F are tangent to. K J quare GHJK is circumscribed about Q. GH, JH, JK, and KG are tangent to Q. ample is circumscribed about. Find the perimeter of. is circumscribed about, so points,, and F are points of tangency. herefore F,, and F. 6 F F 1 1 6 6 8 8 5 1 F 8 he perimeter is 5 units. ercises Find. ssume that segments that appear to be tangent are tangent. 1.. 8 4 square regular heagon 3. 4. 1 6 4 square 5. 1 6. 3 6 4 equilateral triangle Glencoe/McGraw-Hill 566 Glencoe Geometry
10-5 kills ractice angents etermine whether each segment is tangent to the given circle. 1. HI. H 40 I 9 G 41 4 1 13 Find. ssume that segments that appear to be tangent are tangent. 3. 4. 3 6 4 W H Q 10 8 5. F 6. W 8 G 17 4 Z 10 Y Find the perimeter of each polygon for the given information. ssume that segments that appear to be tangent are tangent. 7. Q 4, 9, 13 8. HIJK is a rhombus, I 5, H 13 Q U H V I K U J Lesson 10-5 Glencoe/McGraw-Hill 567 Glencoe Geometry
10-5 ractice angents etermine whether each segment is tangent to the given circle. 1. M. Q M Q 0 1 L 8 48 50 14 Find. ssume that segments that appear to be tangent are tangent. 3. 4. 7 3 L 5 1 U 15 10 Find the perimeter of each polygon for the given information. ssume that segments that appear to be tangent are tangent. 5. 5, U 18, 1 6. KG 3, HG 56 V U L H K G LK For ercises 7 and 8, use the following information. he design shown in the figure is that of a circular clock face inscribed in a triangular base. F and F are equal. 7. Find. in. 1 10 11 1 9 3 8 4 7 6 5 7.5 in. F 8. Find the perimeter of the clock. Glencoe/McGraw-Hill 568 Glencoe Geometry
10-5 eading to Learn Mathematics angents re-ctivity How are tangents related to track and field events? ead the introduction to Lesson 10-5 at the top of page 55 in your tetbook. How is the hammer throw event related to the mathematical concept of a tangent line? eading the Lesson 1. efer to the figure. Name each of the following in the figure. a. two lines that are tangent to Q b. two points of tangency c. two chords of the circle U d. three radii of the circle e. two right angles f. two congruent right triangles g. the hypotenuse or hypotenuses in the two congruent right triangles h. two congruent central angles i. two congruent minor arcs j. an inscribed angle. plain the difference between an inscribed polygon and a circumscribed polygon. Use the words verte and tangent in your eplanation. Helping You emember 3. good way to remember a mathematical term is to relate it to a word or epression that is used in a nonmathematical way. ometimes a word or epression used in nglish is derived from a mathematical term. What does it mean to go off on a tangent, and how is this meaning related to the geometric idea of a tangent line? Lesson 10-5 Glencoe/McGraw-Hill 569 Glencoe Geometry
10-5 nrichment angent ircles wo circles in the same plane are tangent circles if they have eactly one point in common. angent circles with no common interior points are eternally tangent. If tangent circles have common interior points, then they are internally tangent. hree or more circles are mutually tangent if each pair of them are tangent. ternally angent ircles Internally angent ircles 1. Make sketches to show all possible positions of three mutually tangent circles.. Make sketches to show all possible positions of four mutually tangent circles. 3. Make sketches to show all possible positions of five mutually tangent circles. 4. Write a conjecture about the number of possible positions for n mutually tangent circles if n is a whole number greater than four. Glencoe/McGraw-Hill 570 Glencoe Geometry
10-6 tudy Guide and Intervention ecants, angents, and ngle Measures Intersections n or Inside a ircle line that intersects a circle in eactly two points is called a secant. he measures of angles formed by secants and tangents are related to intercepted arcs. If two secants intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle. 1 Q m1 1 (m mq ) If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc. X Y Q U V mxv 1 muv myv 1 mv Lesson 10-6 ample 1 ample Find. he two secants intersect inside the circle, so is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 1 (30 55) 1 (85) 4.5 30 55 Find y. he secant and the tangent intersect at the point of tangency, so the measure the angle is one-half the measure of its intercepted arc. y 1 (168) 84 168 y ercises Find each measure. 1. m1. m 3. m3 5 40 1 9 3 0 U 4. m4 5. m5 6. m6 10 4 V W 90 5 130 160 X 6 Glencoe/McGraw-Hill 571 Glencoe Geometry
10-6 tudy Guide and Intervention (continued) ecants, angents, and ngle Measures Intersections utside a ircle If secants and tangents intersect outside a circle, they form an angle whose measure is related to the intercepted arcs. If two secants, a secant and a tangent, or two tangents intersect in the eterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. Q H G J K N M and are secants. QG is a tangent. QJ is a secant. mgh m 1 (m m ) mq 1 (mgkj ) m 1 M and N are tangents. (mmn mmn ) ample Find mmn. MN is formed by two secants that intersect in the eterior of a circle. mmn 1 (mmn m) M 34 N 18 1 (34 18) 1 (16) or 8 he measure of the angle is 8. ercises Find each measure. 1. m1. m 1 40 80 80 160 3. m3 4. 0 3 70 0 5. 6. 110 50 100 80 Glencoe/McGraw-Hill 57 Glencoe Geometry
10-6 kills ractice ecants, angents, and ngle Measures Find each measure. 1. m1. m 3. m3 50 1 56 48 38 198 Lesson 10-6 3 4. m4 5. m5 6. m6 66 8 6 14 4 50 5 Find. ssume that any segment that appears to be tangent is tangent. 7. 8. 9. 100 7 144 10 40 140 60 10. 11. 1. 45 84 34 Glencoe/McGraw-Hill 573 Glencoe Geometry
10-6 ractice ecants, angents, and ngle Measures Find each measure. 1. m1. m 3. m3 56 16 1 3 146 134 Find. ssume that any segment that appears to be tangent is tangent. 7. 8. 15 9. 101 39 59 6 116 10. 11. 1. 63 5 5 37 9. IN In a game of kickball, ickie has to kick the ball through a semicircular goal to score. If mxz 58 and the mxy 1, at what angle must ickie kick the ball to score? plain. (ball) Z X goal Y Glencoe/McGraw-Hill 574 Glencoe Geometry
10-6 eading to Learn Mathematics ecants, angents, and ngle Measures re-ctivity How is a rainbow formed by segments of a circle? ead the introduction to Lesson 10-6 at the top of page 561 in your tetbook. How would you describe in the figure in your tetbook? When you see a rainbow, where is the sun in relation to the circle of which the rainbow is an arc? Lesson 10-6 eading the Lesson 1. Underline the correct word to form a true statement. a. line can intersect a circle in at most (one/two/three) points. b. line that intersects a circle in eactly two points is called a (tangent/secant/radius). c. line that intersects a circle in eactly one point is called a (tangent/secant/radius). d. very secant of a circle contains a (radius/tangent/chord).. etermine whether each statement is always, sometimes, or never true. a. secant of a circle passes through the center of the circle. b. tangent to a circle passes through the center of the circle. c. secant-secant angle is a central angle of the circle. d. verte of a secant-tangent angle is a point on the circle. e. secant-tangent angle passes through the center of the circle. f. he verte of a tangent-tangent angle is a point on the circle. g. If one side of a secant-tangent angle passes through the center of the circle, the angle is a right angle. h. he measure of a secant-secant angle is one-half the positive difference of the measures of its intercepted arcs. i. he sum of the measures of the arcs intercepted by a tangent-tangent angle is 360. j. he two arcs intercepted by a tangent-tangent angle are congruent. Helping You emember 4. ome students have trouble remembering the difference between a secant and a tangent. What is an easy way to remember which is which? Glencoe/McGraw-Hill 575 Glencoe Geometry
10-6 nrichment rbiting odies he path of the arth s orbit around the sun is elliptical. However, it is often viewed as circular. J H G F Use the drawing above of the arth orbiting the sun to name the line or segment described. hen identify it as a radius, diameter, chord, tangent, or secant of the orbit. 1. the path of an asteroid. the distance between the arth s position in July and the arth s position in ctober 3. the distance between the arth s position in ecember and the arth s position in June 4. the path of a rocket shot toward aturn 5. the path of a sunbeam 6. If a planet has a moon, the moon circles the planet as the planet circles the sun. o visualize the path of the moon, cut two circles from a piece of cardboard, one with a diameter of 4 inches and one with a diameter of 1 inch. ape the larger circle firmly to a piece of paper. oke a pencil point through the smaller circle, close to the edge. oll the small circle around the outside of the large one. he pencil will trace out the path of a moon circling its planet. his kind of curve is called an epicycloid. o see the path of the planet around the sun, poke the pencil through the center of the small circle (the planet), and roll the small circle around the large one (the sun). Glencoe/McGraw-Hill 576 Glencoe Geometry
10-7 tudy Guide and Intervention pecial egments in a ircle egments Intersecting Inside a ircle If two chords intersect in a circle, then the products of the measures of the chords are equal. a c d b a b c d ample Find. he two chords intersect inside the circle, so the products and are equal. 6 8 3 ubstitution 6 4 implify. 4 ivide each side by 6. 3 6 8 Lesson 10-7 ercises Find to the nearest tenth. 1.. 3 6 10 3. 4. 6 8 8 3 7 6 5. 6. 5 7 7. 8. 6 5 3 3 8 6 Glencoe/McGraw-Hill 577 Glencoe Geometry
10-7 tudy Guide and Intervention (continued) pecial egments in a ircle egments Intersecting utside a ircle If secants and tangents intersect outside a circle, then two products are equal. If two secant segments are drawn to a circle from an eterior point, then the product of the measures of one secant segment and its eternal secant segment is equal to the product of the measures of the other secant segment and its eternal secant segment. If a tangent segment and a secant segment are drawn to a circle from an eterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its eternal secant segment. and are secant segments. and are eternal secant segments. Q is a tangent segment. is a secant segment. is an eternal secant segment. () ample Find to the nearest tenth. he tangent segment is, the secant segment is, and the eternal secant segment is. () (18) 15(15 ) 34 5 15 99 15 6.6 18 15 ercises Find to the nearest tenth. ssume segments that appear to be tangent are tangent. 1. 3.3. 3.. 16 18 6 8 V 6 4. 5. 6 6. 13 W 6 Y 3 5 9 4 11 8 7. 35 8. 9. 1 5 15 8 6 Glencoe/McGraw-Hill 578 Glencoe Geometry
10-7 kills ractice pecial egments in a ircle Find to the nearest tenth. ssume that segments that appear to be tangent are tangent. 1.. 3. 3 6 6 7 9 9 18 15 1 Lesson 10-7 4. 5. 4 5 7 9 16 6. 7. 13 5 9 10 8 8. 9. 6 6 1 Glencoe/McGraw-Hill 579 Glencoe Geometry
10-7 ractice pecial egments in a ircle Find to the nearest tenth. ssume that segments that appear to be tangent are tangent. 1.. 3. 5 11 11 8 4 9 1 0 7 4. 5. 8 10 3 15 14 17 6. 7. 5 6 15 6 3 8. 9. 5 0 0 6 10. NUIN n arch over an apartment entrance is 3 feet high and 9 feet wide. Find the radius of the circle containing the arc of the arch. 9 ft 3 ft Glencoe/McGraw-Hill 580 Glencoe Geometry
10-7 eading to Learn Mathematics pecial egments in a ircle re-ctivity How are lengths of intersecting chords related? ead the introduction to Lesson 10-7 at the top of page 569 in your tetbook. What kinds of angles of the circle are formed at the points of the star? What is the sum of the measures of the five angles of the star? eading the Lesson 1. efer to. Name each of the following. a. a diameter b. a chord that is not a diameter F G Lesson 10-7 c. two chords that intersect in the interior of the circle d. an eterior point e. two secant segments that intersect in the eterior of the circle f. a tangent segment g. a right angle h. an eternal secant segment i. a secant-tangent angle with verte on the circle j. an inscribed angle. upply the missing length to complete each equation. a. H H FH b. F c. d. e. F ( ) f. G FG G G F H I Helping You emember 3. ome students find it easier to remember geometric theorems if they restate them in their own words. estate heorem 10.16 in a way that you find easier to remember. Glencoe/McGraw-Hill 581 Glencoe Geometry
10-7 nrichment he Nine-oint ircle he figure below illustrates a surprising fact about triangles and circles. Given any, there is a circle that contains all of the following nine points: (1) the midpoints K, L, and M of the sides of () the points X, Y, and Z, where X, Y, and Z are the altitudes of (3) the points,, and which are the midpoints of the segments H, H, and H that join the vertices of to the point H where the lines containing the altitudes intersect. X Z M K H Y L 1. n a separate sheet of paper, draw an obtuse triangle. Use your straightedge and compass to construct the circle passing through the midpoints of the sides. e careful to make your construction as accurate as possible. oes your circle contain the other si points described above?. In the figure you constructed for ercise 1, draw K, L, and M.What do you observe? Glencoe/McGraw-Hill 58 Glencoe Geometry
10-8 tudy Guide and Intervention quations of ircles quation of a ircle circle is the locus of points in a plane equidistant from a given point. You can use this definition to write an equation of a circle. tandard quation n equation for a circle with center at (h, k) of a ircle and a radius of r units is ( h) (y k) r. y r (h, k) ample Write an equation for a circle with center (1, 3) and radius 6. Use the formula ( h) ( y k) r with h 1, k 3, and r 6. ( h) ( y k) r quation of a circle ( (1)) ( y 3) 6 ubstitution ( 1) ( y 3) 36 implify. ercises Write an equation for each circle. 1. center at (0, 0), r 8. center at (, 3), r 5 Lesson 10-8 3. center at (, 4), r 1 4. center at (1, 4), r 5. center at (, 6), diameter 8 6. center at 1, 1 4, r 3 7. center at the origin, diameter 4 8. center at 1, 5 8, r 5 9. Find the center and radius of a circle with equation y 0. 10. Find the center and radius of a circle with equation ( 4) (y 3) 16. Glencoe/McGraw-Hill 583 Glencoe Geometry
10-8 tudy Guide and Intervention (continued) quations of ircles Graph ircles If you are given an equation of a circle, you can find information to help you graph the circle. ample Graph ( 3) (y 1) 9. Use the parts of the equation to find (h, k) and r. y ( h) ( y k) r ( h) ( 3) ( y k) ( y 1) r 9 h 3 y k y 1 r 3 h 3 k 1 h 3 k 1 (3, 1) he center is at (3, 1) and the radius is 3. Graph the center. Use a compass set at a radius of 3 grid squares to draw the circle. ercises Graph each equation. 1. y 16. ( ) ( y 1) 9 y y (0, 0) (, 1) 3. ( ) y 16 4. ( 1) ( y ) 6.5 y y (, 0) (1, ) 5. 1 y 1 4 4 6. ( y 1) 9 y y ( 1, 1 4 ) (0, 1) Glencoe/McGraw-Hill 584 Glencoe Geometry
10-8 kills ractice quations of ircles Write an equation for each circle. 1. center at origin, r 6. center at (0, 0), r 3. center at (4, 3), r 9 4. center at (7, 1), d 4 5. center at (5, ), r 4 6. center at (6, 8), d 10 7. a circle with center at (8, 4) and a radius with endpoint (0, 4) 8. a circle with center at (, 7) and a radius with endpoint (0, 7) Lesson 10-8 9. a circle with center at (3, 9) and a radius with endpoint (1, 9) 10. a circle whose diameter has endpoints (3, 0) and (3, 0) Graph each equation. 11. y 16 1. ( 1) ( y 4) 9 y y Glencoe/McGraw-Hill 585 Glencoe Geometry
10-8 ractice quations of ircles Write an equation for each circle. 1. center at origin, r 7. center at (0, 0), d 18 3. center at (7, 11), r 8 4. center at (1, 9), d 5. center at (6, 4), r 5 6. center at (3, 0), d 8 7. a circle with center at (5, 3) and a radius with endpoint (, 3) 8. a circle whose diameter has endpoints (4, 6) and (, 6) Graph each equation. 9. y 4 10. ( 3) ( y 3) 9 y y 11. HQUK When an earthquake strikes, it releases seismic waves that travel in concentric circles from the epicenter of the earthquake. eismograph stations monitor seismic activity and record the intensity and duration of earthquakes. uppose a station determines that the epicenter of an earthquake is located about 50 kilometers from the station. If the station is located at the origin, write an equation for the circle that represents a possible epicenter of the earthquake. Glencoe/McGraw-Hill 586 Glencoe Geometry
10-8 eading to Learn Mathematics quations of ircles re-ctivity What kind of equations describe the ripples of a splash? ead the introduction to Lesson 10-8 at the top of page 575 in your tetbook. In a series of concentric circles, what is the same about all the circles, and what is different? eading the Lesson 1. Identify the center and radius of each circle. a. ( ) ( y 3) 16 b. ( 1) ( y 5) 9 c. y 49 d. ( 8) ( y 1) 36 e. ( y 10) 144 f. ( 3) y 5. Write an equation for each circle. a. center at origin, r 8 b. center at (3, 9), r 1 c. center at (5, 6), r 10 d. center at (0, 7), r 7 e. center at (1, 0), d 1 f. center at (4, 8), d g. center at (4.5, 3.5), r 1.5 h. center at (0, 0), r 13 Lesson 10-8 3. Write an equation for each circle. a. y b. y c. y d. y Helping You emember 4. good way to remember a new mathematical formula or equation is to relate it to one you already know. How can you use the istance Formula to help you remember the standard equation of a circle? Glencoe/McGraw-Hill 587 Glencoe Geometry
10-8 nrichment quations of ircles and angents ecall that the circle whose radius is r and whose center has coordinates (h, k) is the graph of ( h) (y k) r.you can use this idea and what you know about circles and tangents to find an equation of the circle that has a given center and is tangent to a given line. (, 3) y y 3 Use the following steps to find an equation for the circle that has center (, 3) and is tangent to the graph y 3. efer to the figure. 1. tate the slope of the line that has equation y 3.. uppose with center (, 3) is tangent to line at point. What is the slope of radius? 3. Find an equation for the line that contains. 4. Use your equation from ercise 3 and the equation y 3. t what point do the lines for these equations intersect? What are its coordinates? 5. Find the measure of radius. 6. Use the coordinate pair (, 3) and your answer for ercise 5 to write an equation for. Glencoe/McGraw-Hill 588 Glencoe Geometry