UI RTI Vocabular heck oncept heck kill heck 2. n two of the points; as shown in ample 3 on p. 81, an two points on a line can be use to calculate the slope of the line. RTI ITIO 1. If ~ X, =, an X =, what is the scale factor of the triangles? 3:2 2. The points (2, 3), (º1, ), (, 1), an (0, ) lie on a line. hich two points coul be use to calculate the slope of the line? plain. ee margin. 3. an ou assume that corresponing sies an corresponing angles of an two similar triangles are congruent? o; corresp. sies of ~ are not unless the scale factor of the is 1:1. orresp. s are, however. etermine whether ~ FH.. no. es 39 72 1 72 F H 9. J an F, an, an H, } J } = } } = F H In the iagram shown J ~. J }} F H. Fin m J, m, an m. 37, 90, 3 10. V an, T an T, 7. Fin an. 10, an U, } V T } = } T T TU } = } V U } 8. iven that, how 11. an Q, an, o ou know that ~? an, } } = } } = plain our answer. Q the Refleive rop. of ong.,, so } Q } ~ b the imilarit ost. 37 0 0 0 0 F 8 J 3 3 H 3 IT UI I a 1: pp. 83 8 s. 9 38 a 2: pp. 8 87 s. 39 2, 7, 1 71 VR a 1: pp. 83 8 s. 9 38 a 2: pp. 8 87 s. 39 3, 7, 1 71 V a 1: pp. 83 8 s. 9 38 a 2: pp. 8 87 s. 39 3, 71 O HU pp. 83 87 s. 9 38 (with 8.3) pp. 8 87 s. 39 3, 7, 1 71 (with 8.) XRI V evel : asier 9 17 evel : ore ifficult 7, 0,, 7 evel : ost ifficult 8, 9,, 8 0 TUT H tra ractice to help ou master skills is on p. 8. TUT H HOOR H ample 1: s. 9 17, 33 38 ample 2: s. 2 ample 3: s. 27 32 ample : s. 39, 3,, ample : s. 7 UI IIRIT TTT The triangles shown are similar. ist all the pairs of congruent angles an write the statement of proportionalit. 9 11. ee margin. 9. 10. V 11. F H J U T OI ROI Use the iagram to complete the following.. QR ~? 13. Q QR = = R ; ;??? 1. 2 0? = 1;? 1 2? 1. = 1 8 2 0? 1. =? 17. =? 1; 1 2 20 R 1 HOOR H To uickl check stuent unerstaning of ke concepts, go over the following eercises: s. 10, 1, 20, 28, 30, 3, 2,, 8. ee also the ail Homework Quiz: lackline aster (hapter 8 Resource ook, p. ) Transparenc (p. 9) 8. imilar Triangles 83 83
OO RROR XRI 23 2 In problems such as these, where the triangles o not have the corresponing parts aligne, it is common for stuents to have trouble ientifing the corresponing parts. To help stuents avoi this ifficult, have them cop the iagram an mark the corresponing angles in the triangles. ome stuents ma also nee to trace one of the triangles an rotate it so that the corresponing parts are more evient. OO RROR XRI 2 In problems such as this, where the triangles are neste an/or rotate, it is common for stuent to have trouble ientifing corresponing parts. To help stuents avoi this ifficult, have them trace both triangles on separate sheets of paper. The can then rotate one of the triangles until the corresponing parts of both triangles align an mark congruent an angles that are share b both triangles in the original figure.. o; m = 31 an m = 7 ; the corresponing Å are not. 22. o; m = 9, m = 9, an m = ; the corresponing Å are not. TRII IIRIT etermine whether the triangles can be prove similar. If the are similar, write a similarit statement. If the are not similar, eplain wh.. 19. R 20. 7 92 21. H F 22. 23. 32 X 8 92 0 77 33 F es; J ~ J es; X ~ FH ee margin. 2. 2. V 2. 0 8 T 0 X R es; ~ es; VX ~ V es; QR ~ T UI R Using the labele points, fin the slope of the line. To verif our answer, choose another pair of points an fin the slope using the new points. ompare the results. 27. 28. 2 ( 8, 3) 1 º} } }3 } ( 3, 1) (2, 1) (7, 3) UI R Fin coorinates for point so that O ~ O. 29. O(0, 0), (0, 3), (, 0), (0, ) (10, 0) 30. O(0, 0), (0, ), (3, 0), (0, 7) } 1 }, 0 31. O(0, 0), (0, 1), (, 0), (0, ) 32. O(0, 0), (0, 8), (, 0), (0, 9) 1 F 7 72 es; QR ~ V 3 33 (30, 0) } 1 }, 0 2 V 9 ( 1, 2) (, 3) 1 R T 1 R T no; }} }}, so the Q lengths of corresponing sies are not proportional. (2, 1) 20 20 2 J (, 0) O (?, 0) 8 hapter 8 imilarit 8
UI R ou are given that is a trapezoi, = 8, =, = 1, an = 10. 33. ~? 3. = 8 =??? ; ; 3.? = 8? 3. 1 = 1 0 } 1 } = } 8 } 1 10 }} = }}?? 10 1 37. =? 20 38. =? TUT H OT Homework Help tuents can fin help for s. 39 at. The information can be printe out for stuents who on t have access to the Internet. TUT H HOOR H Visit our eb site for help with problem solving in s. 39. ITRT IIR TRI The triangles are similar. Fin the value of the variable. 39. 1 8 0. 2} } 7 1 1 r p 1 7 11 OT QUTIO XRI 7 sk stuents to ientif each tpe of special segment shown in these eercises.. : altitue;. : meian;. 7: angle bisector 1. 27 3 2. 3 32 2 z 3. 100. 13} 1 2 } 3 s IIR TRI The segments in blue are special segments in the similar triangles. Fin the value of the variable... 7. 2 80 8 3 8 1 1 20 27 8. ince Æ fi J Æ an J Æ Æ fi, J an J are right Å. ince all right Å are, J J. the Refleive rop. of ong., J J, so J ~ J b the imilarit ost. 8. ROOF rite a paragraph or two-column proof. IV Æ fi Æ J, J Æ Æ fi ROV J ~ J z J 8. imilar Triangles 8 8
THTI ROI XRI 1 sk stuents what the can conclue b showing that this statement is false. ll euilateral triangles are similar. plain that this statement is the negation of the statement given in the eercise. TUT H OT oftware Help Instructions for several software packages are available in blackline format in the hapter 8 Resource ook, p. an at. THI TI XRI tuents will nee some time at school to o this eercise. ou might turn this eercise into a class activit. ivie the class into small groups an give each group a pole of ifferent length. The can then compare their results to see that the height of the builing is inepenent of the length of the pole. 9,. ee itional nswers beginning on page 1. ITIO RTI RTHI For esson 8.: ractice evels,, an (hapter 8 Resource ook, p. ) Reteaching with ractice (hapter 8 Resource ook, p. 8) ee esson 8. of the ersonal tuent Tutor For more ie Review: earch the Test an ractice enerator for ke wors or specific lessons. 1. False; all Å of an 2 euilateral are, so the are ~ b the imilarit ost. (ote also, that if one has sies of length an the other has sies of length, then the ratio of an two sie lengths is } }. Then all corresp. sie lengths are in proportion, so the ef. of ~ can also be use to show that an 2 euilateral are ~.) TUT H OFTR H Visit our eb site to see instructions for several software applications. ITRT 2. True; since the verte of each isosceles has measure 0, the measure of each base is } 1 }(0 º 0 ) = 70. Then 2 the are ~ b the imilarit ost. 8 hapter 8 imilarit 9. ROOF rite a paragraph proof or a two-column proof. The ational Humanities enter is locate in Research Triangle ark in orth arolina. ome of its winows consist of neste right triangles, as shown in the iagram. rove that ~. IV is a right angle, is a right angle. ROV ~ OI ROI In ercises 0 2, ecie whether the statement is true or false. plain our reasoning. 0. If an acute angle of a right triangle is congruent to an acute angle of another right triangle, then the triangles are similar. True; all right Å are, so the are ~ b the imilarit ost. 1. ome euilateral triangles are not similar. 2. ll isosceles triangles with a 0 verte angle are similar. 3. I HO hocke plaer passes the puck to a teammate b bouncing the puck off the wall of the rink as shown. From phsics, the angles that the path of the puck makes with the wall are congruent. How far from the wall will the pass be picke up b his teammate?. THOO Use geometr software to verif that an two points on a line can be use to calculate the slope of the line. raw a line k with a negative slope in a coorinate plane. raw two right triangles of ifferent size whose hpotenuses lie along line k an whose other sies are parallel to the - an -aes. alculate the slope of each triangle b fining the ratio of the vertical sie length to the horizontal sie length. re the slopes eual? es. TH RT RI The reek mathematician Thales (0..) calculate the height of the reat rami in gpt b placing a ro at the tip of the prami s shaow an using similar triangles. In the figure, Q Æ Æ Æ Æ fi QT, R fi QT, an Æ Æ R T. rite a paragraph proof to show that the height of the prami is 80 feet.. TITI HIHT On a sunn a, use a ro or pole to estimate the height of our school builing. Use the metho that Thales use to estimate the height of the reat rami in ercise. nswers will var. 1. m ee margin. 1 m 2. m ee margin. puck m 780 ft ee margin. wall ot rawn to scale R. ft ft T 8
Test 7. UTI-T RO Use the following information. reparation 7. a. Right Å X an X are oing from his own house to Raul s house, ark rives ue south one mile, ue east three miles, an vertical Å X an ue south again three miles. hat is the 1 mi an X are. Then istance between the two houses as the crow flies? X ~ X b the imilarit ost. a. plain how to prove that X ~ X. hallenge XTR H IX RVI b. Use corresponing sie lengths of the triangles to calculate X. } 3 } mi c. Use the thagorean Theorem to calculate X, an then X. Then fin. 1.2 mi; 3.7 mi; mi Raul s house ark s house. riting Using the properties of rectangles, eplain a wa that a point coul be ae to the iagram so that Æ woul be the hpotenuse of Æ Æ, an an woul be its legs of known length. ee margin. HU VIIO In ercises 8 0, use the following information. The iagram shows how similar triangles relate to human vision. n image similar to a viewe object appears on the retina. The actual height of the object h is proportional to the size of the image as it appears on the retina r. In the same manner, the istances from the object to the lens of the ee an from the lens to the retina, 2 mm in the iagram, are also proportional. 8. rite a proportion that relates r,, h, an 2 mm. ample answer: } 2 } = }h r } 9. n object that is 10 meters awa appears on the retina as 1 mm 2 mm tall. Fin the height of the object. 00 mm lens 0. n object that is 1 meter tall h r appears on the retina as 1 mm tall. How far awa is the object? ot rawn to scale retina 2 m 7.. ample answer: onstruct a line to (º17, ) an (1, º21). (Review 1.3) 82 1. UI TH IT FORU Fin the istance between the points Æ through an eten Æ TRI IT,, an up to this line to intersect it at. ince are the mipoints of the sies of J. Æ Æ Æ Æ omplete the statement.,, (Review. for 8.) an is a right, J is a rectangle. Æ 2. Æ Thus = = 1 mi? J 3. If = 23, then J =?. an = = 3 mi. ince is a right,. If = 1, then =?. 1. If J = 2, then =?. is a right triangle Æ ROORTIO olve the proportion. (Review 8.1) with hpotenuse an legs of 1 mi an. = 3 1 2 8 7. 3 = 1 2 32 8. 1 7 = 1 1 3 mi. } 9 2 } 8 33 1 9. 3 11 = + 70. 2 3 3 2 = 71. 8 72 = } 3 } 9 º1, 1 11 32 X 3 mi 3 mi I HOOR QUI Transparenc vailable Use the iagram for ercises 1. 7. U V X 1. UV X 2. hat is the scale factor of UV to X? } } 3. hat is V? 10. hat is X? 9. If m U = 0 an m = 30, what is m? 100 XTR H OT hallenge problems for esson 8. are available in blackline format in the hapter 8 Resource ook, p. 2 an at. ITIO TT RRTIO 1. RITI plain how to use shaows to inirectl measure the height of a tall object. Illustrate our eplanation with a sketch. ample answer: rect a pole perpenicular to the groun. easure the height of the pole an the length of its shaow. Then measure the length of the shaow of the tall object. Use similar triangles to fin the height of the tall object. heck stuents sketches. 8. imilar Triangles 87 87