Angles. Chapter Segments and

Transcription

1 Chapter Segments and nges H w tr k d t r d? In baseba, the strike zone is the area above home pate where a ba is considered a strike. The ocation of the strike zone is based on each payer s height and batting stance. Th id i b h f h b s h d d h f h b s b d i h f h ik L b I You wi earn more about strike zones in Eercises 36 and 37 on p

2 Who uses Segments and nges? KITE DESIGNER Kite designers use geometric principes in designing and making kites. kite s struts often bisect the anges they support. (p. 6) ERGONOIST Ergonomists design offices, furniture, and equipment to improve the safety and comfort of workers. For eampe, the work space shown at the right was designed to enhance privacy in offices with an open foor pan. (p. 80) How wi you use these ideas? Determine the strike zone for different batting stances. (p. 58) Describe ange reationships in a paper airpane pattern. (p. 65) redict the ange of refection of a aser. (p. 65) Learn about ange reationships found in the amio ridge in Spain. (p. 7) Interpret the meaning of advertising sogans. (p. 86) 5

3 Chapter Study Guide REVIEW What th h pter b ut? nayzing segment bisectors and ange bisectors Identifying compementary anges, suppementary anges, vertica anges, and inear pairs Using properties of equaity and congruence Key Words midpoint, p. 53 segment bisector, p. 53 bisect, p. 53 ange bisector, p. 6 compementary anges, p. 67 suppementary anges, p. 67 adjacent anges, p. 68 theorem, p. 69 vertica anges, p. 75 inear pair, p. 75 if-then statement, p. 8 deductive reasoning, p. 83 RERE Chapter Read s Q iz Take this quick quiz. If you are unsure of an answer, ook at the reference pages for hep. Vocabuary Check (refer to p. 36). Suppose mac 00. What type of ange is ac? acute right C obtuse D straight Ski Check (refer to pp. 37, 67). In the diagram, maqr 65 and maqs. What is masqr? F G 43 H 53 J Which of the foowing is a soution of the equation 4 0? 80 6 C 5 D 5 S R VISUL STRTEGY Visuaize It! icturing Theorems In this chapter, you wi earn the first of many theorems. To hep you visuaize a theorem, draw an eampe that uses specific measures. Vertica nges Theorem Vertica anges are congruent. 5 Chapter Segments and nges

4 . Segment isectors Goa isect a segment. Find the coordinates of the midpoint of a segment. Key Words midpoint segment bisector bisect Geo-ctivity On a piece of paper, draw &*. Foding a Segment isector Fod the paper so that point is on top of point. 3 Labe the point where the fod intersects &* as point. 4 Use a ruer to measure the engths of &**, &*, and &*. What do you notice? VOCULRY TI i- means two, and -sect means to cut. So, bisect means to cut in two. In the Geo-ctivity, is caed the midpoint of &*. The midpoint of a segment is the point on the segment that divides it into two congruent segments. segment bisector is a segment, ray, ine, or pane that intersects a segment at its midpoint. To bisect a segment means to divide the segment into two congruent segments. D C is the midpoint of &*. CD is a bisector of &*.. Segment isectors 53

5 STUDY TI The midpoint of a segment divides the segment in haf. If you know the ength of &*, mutipy by to find and. EXLE Find Segment Lengths is the midpoint of &*. Find and. Soution is the midpoint of &*, so and are each haf the ength of &*. p p 6 3 NSWER 3 and 3. 6 EXLE Find Segment Lengths is the midpoint of RS&*. Find S and RS. R 7 S Soution is the midpoint of RS&*, so S R. Therefore, S 7. You know that RS is twice the ength of R &*. RS p R p 7 4 NSWER S 7 and RS 4. Find Segment Lengths. Find DE and EF.. Find N and. 8 D E F N EXLE 3 Use gebra with Segment Lengths Line is a segment bisector of &*. Find the vaue of. Soution 5 35 Line bisects &* at point Substitute 5 for and 35 for Divide each side by Simpify. CHECK Check your soution by substituting 7 for. 5 5(7) Chapter Segments and nges

6 idpoints If you know the coordinates of the endpoints of a ine segment in a coordinate pane, you can find the coordinates of the midpoint of the segment using the idpoint Formua. THE IDOINT FORUL REDING TI The numbers and in and y are caed subscripts. You read as sub and y as y sub. Words The coordinates of the midpoint of a segment are the averages of the -coordinates and the y-coordinates of the endpoints. Symbos The midpoint of the segment joining (, y ) and (, y ) y y y y y (, y ) (, y ) y, y y is, y. EXLE 4 Use the idpoint Formua Find the coordinates of the midpoint of &*. a. (, ), (7, 4) b. (, 3), (5, ) Soution First make a sketch. Then use the idpoint Formua. SKILLS REVIEW For hep potting points in a coordinate pane, see p a. y b. (7, 4) (, ) (, 3) y (5, ) Let (, y ) (, ) Let (, y ) (, 3) and (, y ) (7, 4). and (, y ) (5, )., y 7, 4 (4, 3) y 3,, y y 5, 3 ( ) Use the idpoint Formua Sketch Q&*. Then find the coordinates of its midpoint. 3. (, 5), Q(4, 3) 4. (0, ), Q(4, 0) 5. (, ), Q( 4, ). Segment isectors 55

7 . Eercises Guided ractice Vocabuary Check. In the diagram shown at the right, name the midpoint and a segment bisector of &*. C Ski Check is the midpoint of the segment. Find the segment engths.. Find R and S. 3. Find F and G R S F 4. Find Q and Q. 5. Find Y and YZ. G 4 Y 0 Z is the midpoint of JK&*. Find the vaue of the variabe J 4 33 K J 6r 48 K Find the coordinates of the midpoint of R&*. 8. y 9. y 0. (, ) (, 3) R(4, ) R (, 3) ( 3, 0) 4 y R (5, 6) ractice and ppications Etra ractice See p Recognizing idpoints In Eercises 4, determine whether is the midpoint of &*. Epain your reasoning Visuaize It! Sketch a ine segment, Q&*, that is bisected by ine at point R. 56 Chapter Segments and nges

8 Homework Hep Eampe : Es. 6 9 Eampe : Es. 0 3 Eampe 3: Es. 6 9 Eampe 4: Es Finding Segment Lengths is the midpoint of the segment. Find the segment engths. 6. Find K and L. 7. Find D and E. K 38 L D 8 E 8. Find Y and Z. 9. Find and. 7.7 Y Z Finding Segment Lengths Line bisects the segment. Find the segment engths. 0. Find C and.. Find and N. 36 C 5 N. Find FE and DE. 3. Find UT and ST. D 9.5 F E S 3.6 U T iking The inuteman ikeway is a 0.5 mie bike path that runs from rington to edford, assachusetts. 4. Caitin and Laurie begin at opposite ends of the inuteman ikeway and meet at the hafway point on the path. How far does each rider bike? 5. Caitin starts on the path 4.3 mies from the rington end. Laurie starts on the path 3 mies from the edford end. How far wi each rider bike before reaching the hafway point on the path? Using gebra Find the vaue of the variabe p 7 9 q 7 I CLSSZONE.CO r s 6 HOEWORK HEL Etra hep with probem soving in Es is at casszone.com idpoint Formua Find the coordinates of the midpoint of Q&*. 30. (0, 0), Q(4, 6) 3. (3, 8), Q(7, 6) 3. ( 5, 6), Q(9, 7) 33. (, 0), Q(6, ) 34. ( 4, 4), Q(4, 0) 35. (3, ), Q( 7, 4). Segment isectors 57

9 LOOK CK For more about baseba, see p. 50. Strike Zone In Eercises 36 and 37, use the information beow. In baseba, the strike zone is the region a baseba needs to pass through for the umpire to decare it a strike if the batter does not swing. The top of the strike zone is a horizonta pane passing through the midpoint of the top of the batter s shouders and the top of the uniform pants when the payer is in a batting stance. Source: ajor League aseba 36. Find the coordinate of T. 37. Find the coordinate of T. 60 T 4 63 T VOCULRY TI Lines of atitude run parae to the Equator. Lines of ongitude run north-south. EXLE Latitude-Longitude Coordinates Find the coordinates of the pace hafway between San Francisco (37.8 N,.4 W) and Los ngees (34. N, 8. W). Soution, y y , N 36 N W 0 W San Francisco (37.8 N,.4 W) 8 W CLIFORNI NEVD 6 W (35.95, 0.3 W) 34 N Los ngees (34. N, 8. W) NSWER The pace hafway between San Francisco and Los ngees has coordinates (35.95 N, 0.3 W). 58 Chapter Segments and nges Latitude-Longitude Coordinates Find the coordinates of the pace hafway between the two cities in Caifornia. 38. Fresno: (36.7 N, 9.8 W) 39. ishop: (37.4 N, 8.4 W) Napa: (38.3 N,.3 W) Los ngees: (34. N, 8. W) 40. San Francisco: (37.8 N,.4 W) 4. Santa arbara: (34.4 N, 9.7 W) ao to: (37.4 N,. W) Oakand: (37.8 N,.3 W)

10 4. Using idpoints In the diagram beow, is the midpoint of C&*, 9, and D 5. Find CD. 5 9 C D 43. Chaenge The midpoint of &* is (7, 5). The coordinates of point are (4, ). Find the coordinates of point. Epain. Standardized Test ractice 44. utipe Choice T is the midpoint of QR &*. What is the vaue of? 7 C 9.5 D 88 T R 45. utipe Choice What is the midpoint of the segment joining (, 7) and ( 6, )? F, 9 G ( 4, 9) H (, 4) J 9, ied Review Evauating Statements Use the diagram at the right to determine whether the statement is true or fase. (Lessons.3,.5) 46. oint ies on ine m. 47. oint E ies on ine. 48. oints, E, and C are coinear. 49. Lines and m intersect at point E. 50. oint E is between points and C. 5. oint F is between points and. D E C F m Cassifying nges Name the verte and sides of the ange. Then state whether it appears to be acute, right, obtuse, or straight. (Lesson.6) J K L R C gebra Skis Evauating Epressions Evauate the epression. (Skis Review, p. 670) 55. p p p p p p p 3 p p (3 4) p 5. Segment isectors 59

11 ctivity. Foding nge isectors Question How can you bisect an ange? aterias protractor straightedge Epore On a piece of paper, use a straightedge to draw an acute ange. Labe the ange ac. Fod the paper so C &*( is on top of &*(. 3 Draw a point D on the fod inside ac. Then use a protractor to measure ad, adc, and ac. LOOK CK For hep using a protractor, see p. 36. Think bout It. What do you notice about the anges you measured in Step 3?. Repeat Steps through 3 with an obtuse ange. Compare your resuts with the resuts from Eercise. 3. Copy and compete: a. mad? p mac b. madc? p mac c. mac? p mad d. mac? p madc 4. Etension Is it possibe to fod congruent anges from a straight ange? Epain your reasoning. 60 Chapter Segments and nges