hpter17 ongruene nd trnsfortions ontents: Trnsfortions ongruent figures ongruent tringles Proof using ongruene
352 ONGRUENE N TRNSFORMTIONS (hpter 17) Opening prole Jne ut two tringulr slies of heeseke, nd gve one to her rother Nthn. Tht s not fir, Nthn sid, our slie is igger thn ine. Jne used ruler to esure the sides of eh slie. See, slie hs sides 5, 6, nd 7, nd so does ours. Tht ens the slies re the se size. Not neessril, sid Nthn, the slies ight hve the se sides, ut the ngles ight e different. Things to think out: Who do ou think is orret? Wht thetil rguent n ou use to justif our nswer? ongruene is rnh of geoetr tht dels with ojets whih re identil in size nd shpe. In this hpter we will review trnsfortions, nd look t how we n use trnsfortions to define ongruene. We will then use ongruene to prove the properties of polgons we hve studied erlier in the er. TRNSFORMTIONS trnsfortion is proess whih hnges either the size, shpe, orienttion, or position of figure. When we perfor trnsfortion, the originl figure is lled the ojet, nd the resulting figure is lled the ige. In this setion we revise the trnsltion, refletion, nd rottion trnsfortions. TRNSLTIONS trnsltion is trnsfortion in whih ever point on the figure oves fied distne in given diretion. 4 This ojet hs een trnslted 4 units right nd 3 units down. ojet ige 3 Under trnsltion, the size nd shpe of n ojet does not hnge. Onl the position of the ojet hnges.
ONGRUENE N TRNSFORMTIONS (hpter 17) 353 Eple 1 Trnslte this ojet 5 units left nd 2 units up. Self Tutor 0 is the ige of the ojet. 0 2 5 We n lso trnslte ojets plotted on rtesin plne. Eple 2 Trnslte the qudrilterl 2 units right nd 4 units up. Stte the verte oordintes of the ige qudrilterl. Self Tutor The verties of the ige qudrilterl 0 re 0 ( 1, 3), 0 (1, 1), 0 (1, 1), nd 0 ( 1, 1). 0 0 2 4 0 EXERISE 17.1 1 Trnslte the given figures in the diretion indited: PRINTLE IGRMS 3 units right, 2 units down. 4 units right. 3 units down.
354 ONGRUENE N TRNSFORMTIONS (hpter 17) d e f 3 units left, 1 unit up. 1 unit left, 4 units down. 2 eterine the trnsltion fro to 0 in the following: 3 units left, 3 units up. 0 0 0 d e f 0 0 0 3 n ojet is trnslted 5 units right nd 3 units down to 0. esrie the trnsltion fro 0 k to. 4 Trnslte the following figures in the diretion given, nd stte the verte oordintes of the ige: F E Trnslte 4 units left. Trnslte 2 units right, then 1 unit down. Trnslte 3 units left, then 4 units up.
ONGRUENE N TRNSFORMTIONS (hpter 17) 355 Eple 3 Self Tutor Is this trnsfortion trnsltion? 0 0 0 In trnsltion, ever point on the figure oves the se distne in the se diretion. However, hs oved 3 units right nd 1 unit up, while hs oved 4 units right nd 1 unit up. 0 0 0 The tringles do not hve the se shpe, so it nnot e trnsltion. So, this trnsfortion is not trnsltion. 5 re the following trnsfortions trnsltions? If so, desrie the trnsltion. 0 0 0 0 0 0 0 0 0 0 0 0 F E F 0 E 0 6 onsider the figures longside. Whih of these figures is trnsltion of figure? esrie the trnsltion fro figure to this figure. Whih of these figures is trnsltion of figure G? esrie the trnsltion fro figure G to this figure. Whih of these figures nnot e trnslted to n other figure? H I G E F
356 ONGRUENE N TRNSFORMTIONS (hpter 17) REFLETIONS onsider the figure longside. The ojet hs een refleted in the irror line to for its ige. In this se we ight ll it the irror ige. ojet ige irror line Investigtion 1 Refletions You will need: Wht to do: irror, pper, penil, ruler. PRINTLE FIGURES 1 Mke two opies of the figures shown elow: d 2 Put the irror long the irror line on one op. Wht do ou notie in the irror? 3 rw the refletion s urtel s ou n on the seond op. 4 ut out the seond op with its refletion nd fold it long the irror line. You should find tht the two prts of the figure n e folded etl onto one nother long the irror line. When point is refleted in irror line, nd its ige 0 re the se distne fro the irror line, nd the line joining nd 0 is perpendiulr to the irror line. 0 irror line
ONGRUENE N TRNSFORMTIONS (hpter 17) 357 Eple 4 Self Tutor Reflet the following figures in the given irror lines: 0 0 EXERISE 17.2 1 op the following figures onto grid pper nd reflet the in the given irror lines: PRINTLE IGRMS d e f Eple 5 Self Tutor Reflet this figure in the -is. 0
358 ONGRUENE N TRNSFORMTIONS (hpter 17) 2 Reflet these figures in the -is: 3 Reflet these figures in the -is: 4 qudrilterl hs verties (1, 5), (4, 3), (4, 1), nd (1, 1). Plot the qudrilterl on rtesin plne. Reflet in the -is, nd stte the verte oordintes of the ige. Reflet in the -is, nd stte the verte oordintes of the ige. 5 op nd oplete: When the point (, ) is refleted in the -is, the ige hs oordintes (::::, ::::). When the point (, ) is refleted in the -is, the ige hs oordintes (::::, ::::). 6 For eh of the following, deterine whether 0 is refletion of in the -is: 0 0 0 7 Whih two of the figures re refletions of eh other? Whih of the es is the irror line for this refletion? Whih two of the figures re trnsltions of eh other?
ROTTIONS ONGRUENE N TRNSFORMTIONS (hpter 17) 359 When wheel oves out its le, we s tht the wheel rottes. The entre point on the le is the entre of rottion. The ngle through whih the wheel turns is the ngle of rottion. Other eples of rottion re the oveent of the hnds of lok, nd opening nd losing door. rottion is trnsfortion in whih ever point on the figure is turned through given ngle out fied point. The fied point is lled the entre of rottion nd is usull lelled O. For eple, the ojet longside hs een rotted 90 ± ntilokwise out O. When point is rotted out O, tht point nd its ige re the se distne fro O. O = O 0, O = O 0, nd so on. We drw irle rs entred t O to ke sure tht point nd its ige re the se distne fro O. 0 0 0 ige 0 E 0 ojet E 90 O Eple 6 Self Tutor Rotte the given figures out O through the ngle indited: O O O 180 ± lokwise 90 ± ntilokwise 90 ± lokwise O O O
360 ONGRUENE N TRNSFORMTIONS (hpter 17) EXERISE 17.3 1 erek is sitting in hir. Whih hir will he ove to if he rottes ntilokwise out O through n ngle of: 90 ± 270 ± 180 ±? O 2 Rotte the given figures out O through the ngle indited: O O O 90 ± ntilokwise 180 ± 90 ± lokwise We rotte ntilokwise unless we re told otherwise. Eple 7 Self Tutor Stte the verte oordintes of tringle. Rotte the tringle 90 ± lokwise out the origin O. Stte the verte oordintes of the ige. The tringle hs verties ( 2, 4), ( 1, 4), nd ( 2, 1). The ige tringle hs verties 0 (4, 2), 0 (4, 1), nd 0 (1, 2). 0 0 0
ONGRUENE N TRNSFORMTIONS (hpter 17) 361 3 Stte the verte oordintes of tringle PQR. Rotte the tringle 90 ± ntilokwise out the origin O. Stte the verte oordintes of the ige. P Q R 4 qudrilterl hs verties ( 4, 3), ( 1, 3), ( 1, 4), nd ( 4, 4). Plot on rtesin plne. Rotte 90 ± lokwise out the origin O. Stte the verte oordintes of the ige. 5 Whih of the figures longside is rottion of out the origin? eterine the ngle of rottion fro figure to this figure. isussion When n ojet is trnslted, refleted, or rotted, does the size of the ojet hnge? oes the shpe of the ojet hnge? ONGRUENT FIGURES Two figures re ongruent if the re identil in size nd shpe. The do not need to hve the se orienttion. For eple, the figures longside re ongruent even though one is rottion of the other. ongruent figures
362 ONGRUENE N TRNSFORMTIONS (hpter 17) tivit reting ongruent figures You will need: Two sheets of pper, sissors. Wht to do: 1 rw shpe on one of the sheets of pper. 2 Ple the seond sheet of pper ehind it, nd hold the together tightl. refull ut out the shpe, utting through oth sheets of pper. This will give ou two ongruent figures. 3 In group or s lss, ple oth figures fro eh student in o, nd i the figures up. 4 Tr to pir up the ongruent figures. How n ou tell tht two figures re ongruent? The figures longside re ongruent. The orresponding sides nd ngles in the figures re identil. If we were to ple one figure on top of the other, the would th eh other perfetl. EMO EMO Eple 8 Self Tutor re the following pirs of figures ongruent? The figures do not hve the se shpe, so the re not ongruent. The figures re identil in size nd shpe even though one is rotted. The re therefore ongruent. lthough the figures hve the se shpe, the re not the se size. The re not ongruent.
ONGRUENE N TRNSFORMTIONS (hpter 17) 363 EXERISE 17.1 1 re the following pirs of figures ongruent? d 2 Whih two of these figures re ongruent? E 3 Whih three of these figures re ongruent? E 4 Qudrilterls EFGH nd re ongruent. eterine the: length of side [EF] size of ngle FGH perieter of EFGH. 10 65 107 12 7.8 90 98 8 G F H E USING TRNSFORMTIONS TO EFINE ONGRUENE The figures nd longside re ongruent. If we trnslte figure to figure, the two figures fit together perfetl. EMO
364 ONGRUENE N TRNSFORMTIONS (hpter 17) The figures P nd Q re ongruent. The hve different orienttions, ut the still hve the se size nd shpe. P Q The orienttion of figure refers to the diretion it is fing. We n see this rotting P so tht it hs the se orienttion s Q, then trnslting the ige P 0 to Q. EMO P 0 Q The figures X nd Y re lso ongruent. To see tht the re the se size nd shpe, we reflet figure X in irror line, then trnslte the ige X 0 to Y. X Y X Y EMO X 0 We n therefore use trnsfortions to define ongruene: Two figures re ongruent if one figure lies etl on top of the other fter ointion of trnsltions, rottions, nd refletions. Eple 9 Show tht nd re ongruent trnsforing onto. Self Tutor We first reflet figure in the -is. We then trnslte 0 5 units right nd 1 unit down. 0 The ige fits onto figure etl, so nd re ongruent.
ONGRUENE N TRNSFORMTIONS (hpter 17) 365 EXERISE 17.2 1 Show tht is ongruent to trnsforing to in single trnsfortion: d e f 2 Show tht is ongruent to trnsforing to in ointion of trnsfortions: d e f PRINTLE IGRMS First reflet or rotte so the figures hve the se orienttion. Then trnslte if required.
366 ONGRUENE N TRNSFORMTIONS (hpter 17) 3 re the following pirs of figures ongruent? ONGRUENT TRINGLES The tringles longside hve identil side lengths nd ngles, so the tringles re ongruent. However, we do not need to know ll of the infortion given to onlude tht the tringles re ongruent. 5 82 7 60 38 8 5 60 82 8 38 7 For eple, s the investigtion elow deonstrtes, knowing tht the tringles hve the se side lengths is suffiient to onlude tht the tringles re ongruent. Investigtion 2 ongruent tringles Wht to do: 1 lik on the ion to run the oputer deonstrtion. The oputer will generte tringle with side lengths 8, 10, nd 12. You will now ke nother tringle with these diensions. 2 hoose whih of the side lengths ou would like to strt with. For eple, ou hoose to strt with the 12 side. 3 rg the third verte of the tringle round until the reining two sides hve the orret lengths. 8 12 10 EMO 12 4 opre our tringle with the one generted the oputer. Wth the trnsfortions nd deide if the tringles re ongruent. 5 onstrut nother tringle with side lengths 8, 10, nd 12, nd test this tringle for ongruene with the other two.
You should hve found tht our tringles nd the oputer s tringles were ongruent. If we know tht two tringles hve the se side lengths, then these tringles ust e ongruent. ONGRUENE N TRNSFORMTIONS (hpter 17) 367 Investigtion 3 onstruting tringles In this investigtion we will disover other onditions whih llow us to onlude tht two tringles re ongruent. You will need: Pper, ruler, protrtor. Wht to do: 1 Two sides nd n inluded ngle rw tringle with two side lengths 8 nd 12, with n ngle of 25 ± etween these sides. How n different tringles n e onstruted? 2 Two ngles nd orresponding side rw tringle with two ngles esuring 70 ± nd 45 ±, with the side etween these ngles eing 10 long. How n different tringles n e onstruted? 3 Right ngle, hpotenuse, nd side rw right ngled tringle with hpotenuse 10, nd one other side 6 long. How n different tringles n e onstruted? 4 Two sides nd non-inluded ngle rw tringle with two side lengths 8 nd 12, with n ngle of 25 ± etween the 12 side nd the third side s shown. How n different tringles n e onstruted? 5 Three ngles rw tringle with ngles 50 ±, 60 ±, nd 70 ±. How n different tringles n e onstruted? 6 8 25 12 70 45 10 The hpotenuse is the longest side of right ngled tringle. 8 12 70 10 25 60 50
368 ONGRUENE N TRNSFORMTIONS (hpter 17) You should hve de the following disoveries: Two tringles re ongruent if n one of the following is true: ² ll orresponding sides re equl in length. (SSS) ² Two sides nd the inluded ngle re equl. (SS) ² Two ngles nd pir of orresponding sides re equl. (ors) ² For right ngled tringles, the hpotenuses nd one pir of sides re equl. (RHS) We usull indite our reson wh two tringles re ongruent writing one of the revitions given ove in old. If we know two side lengths nd non-inluded ngle, there e two ws to onstrut the tringle. This is therefore not suffiient infortion to show tht two tringles re ongruent. If we know ll ngles of tringle, the tringle still vr in size. This is therefore not suffiient infortion to show tht two tringles re ongruent. Eple 10 Self Tutor re these pirs of tringles ongruent? Give resons for our nswers. d 4 4 Yes frhsg Yes fssg No. This is not ors s the equl sides re not in orresponding positions. One is opposite ngle, the other is opposite ngle. d Yes fors g
ONGRUENE N TRNSFORMTIONS (hpter 17) 369 One we hve estlished tht two tringles re ongruent, we n dedue tht the reining orresponding sides nd ngles of the tringles re equl. Eple 11 Self Tutor onsider the two tringles longside. Show tht the tringles re ongruent. Wht n e dedued fro this ongruene? X Y Z»= ens is ongruent to = XY, = YZ nd = XYZ So, 4» = 4XYZ fssg X Y nd = XZ = YXZ = XZY Z When we desrie ongruent tringles, we lel the verties tht re in orresponding positions in the se order. For instne, in the previous eple, we write 4» = 4XYZ, not 4» = 4YZX. EXERISE 17 1 Stte whether these pirs of tringles re ongruent, giving resons for our nswers: d e f 30 60 30
370 ONGRUENE N TRNSFORMTIONS (hpter 17) 2 Stte whether these pirs of tringles re ongruent, giving resons for our nswers: d e f 100 30 50 50 3 Whih of the following tringles is ongruent to the one longside? 8 80 5 8 8 80 5 8 80 70 5 8 5 5 80 4 Whih of these tringles re ongruent to eh other? 15 15 17 17 50 9 E F 40 15 9 50 40 15 17 9
isussion ONGRUENE N TRNSFORMTIONS (hpter 17) 371 5 i ii d e f g h We hve seen tht if two tringles hve equl orresponding sides, then the re ongruent. Is the se true for qudrilterls? n we s tht the qudrilterls longside re ongruent? Q P R J K L X Z Y F P Q R E E P T R S Q T R S Z X Y For eh of the following pirs of tringles, whih re not drwn to sle: eterine whether the tringles re ongruent. If the tringles re ongruent, wht else n we dedue out the? F E E F X Y W
372 ONGRUENE N TRNSFORMTIONS (hpter 17) PROOF USING ONGRUENE In hpter 8, we studied the properties of isoseles tringles nd speil qudrilterls. We n use ongruene to prove n of these properties. Eple 12 onsider the isoseles tringle. M is the idpoint of []. Use ongruene to show tht M = M. Wht propert of isoseles tringles hs een proven? Self Tutor M In tringles M nd M: ² = f4 is isoselesg ² M = M fm is the idpoint of []g ² [M] is oon to oth tringles. ) 4M» = 4M fsssg Equting orresponding ngles, M = M. In n isoseles tringle, the line joining the pe to the idpoint of the se isets the vertil ngle. EXERISE 17 1 onsider the prllelogr. op nd oplete: In tringles nd : ² =... flternte nglesg ² =... flternte nglesg ² [] is oon to oth tringles ) 4» = 4 f...g Equting orresponding ngles, =... Wht propert of prllelogrs hs een proven in? 2 onsider the kite PQRS. Use ongruene to show tht QPR = SPR nd QRP = SRP. Wht propert of kites hs een proven? P Q S R
ONGRUENE N TRNSFORMTIONS (hpter 17) 373 3 onsider the squre. Show tht 4» = 4. Hene, show tht =. Wht propert of squres hs een proven? 4 X Y onsider the rhous WXYZ. Show tht 4WXY» = 4YZW. W Z Hene, show tht XYW = Z WY. Eplin wh [XY] is prllel to [WZ]. d e Likewise, show tht [XW] is prllel to [YZ]. Wht propert of rhouses hs een proven? 5 The digonls of rhous PQRS eet t M. Show tht 4PSQ» = 4RSQ. Hene, show tht PSQ = RSQ. Wht propert of rhouses hs een proven? d Eplin wh 4PSM» = 4RSM. e Hene: i show tht PM = RM ii find the sizes of S MP nd S MR. P Q M S R f Use e to show tht 4SMP» = 4QMR, nd therefore SM = QM. g Wht propert of rhouses hs een proven in e nd f? 6 Use ongruene to show tht: the opposite sides of prllelogr re equl in length the se ngles of n isoseles tringle re equl the digonls of kite interset t right ngles. Review set 17 1 Trnslte the given figures in the diretion indited: 3 units right 2 units left nd 2 units down
374 ONGRUENE N TRNSFORMTIONS (hpter 17) 2 esrie the following trnsfortions: 0 0 0 0 0 0 0 0 E F F0 E0 3 Rotte eh figure out O through the ngle indited: O O 180 ± 4 Reflet eh figure in the is indited: 270 ± ntilokwise -is 5 Stte the verte oordintes of the qudrilterl PQRS longside. Rotte the qudrilterl 90 ± ntilokwise out the origin O. Stte the verte oordintes of the ige. -is 3 Q 4 4 R P S 3 6 Whih two of the figures longside re trnsltions of eh other? Whih two of the figures re refletions of eh other? Whih of the es is the irror line for this refletion?
ONGRUENE N TRNSFORMTIONS (hpter 17) 375 7 Show tht is ongruent to trnsforing to in single trnsfortion: 8 onsider the kite. Use ongruene to show tht =. Wht propert of kites hs een proven? 9 For eh of the following pirs of tringles, not drwn to sle: i eterine whether the tringles re ongruent. ii If the tringles re ongruent, wht n e dedued fro the ongruene? Y T X Z S U F E 10 Show tht is ongruent to trnsforing to in ointion of trnsfortions: 11 Use ongruene to prove tht, in n isoseles tringle, the line joining the pe to the idpoint of the se eets the se t right ngles. Prtie test 17 lik on the link to otin printle version of this test. Multiple hoie PRINTLE TEST
376 ONGRUENE N TRNSFORMTIONS (hpter 17) Prtie test 17 Short response 1 R S esrie the trnsltion fro R to S. 2 Trnslte the qudrilterl 2 units right nd 3 units down. Stte the verte oordintes of the ige. op the given figure nd trnslte it 5 units left nd 2 units up. 3 op the figure nd reflet it in the irror line shown. 4 Show tht P is ongruent to Q single trnsfortion of P to Q. P Q Q P 5 tringle hs verties (2, 1), (4, 3), nd (3, 0). Plot tringle on rtesin plne. Reflet 4 in the -is, nd stte the verte oordintes of the ige. Reflet 4 in the -is, nd stte the verte oordintes of the ige. 6 Stte whether these pirs of tringles re ongruent, giving resons for our nswers.
ONGRUENE N TRNSFORMTIONS (hpter 17) 377 7 Whih of the figures longside is rottion of out the origin? eterine the ngle of rottion fro figure to this figure. 8 Stte whether eh pir of figures is ongruent, giving resons for our nswer: 9 Show tht is ongruent to trnsforing to in ointion of trnsfortions: 10 Stte whether these pirs of tringles re ongruent, giving resons for our nswers: Prtie test 17 Etended response 1 Tringle T hs oordintes ( 2, 1), (1, 3), nd (2, 1). Plot the tringle T on rtesin plne. Trnslte T 4 units right nd 2 units up. Stte the verte oordintes of the ige tringle T 0. Trnslte T 0 1 unit left nd 5 units down. Stte the verte oordintes of the ige tringle T 00. d esrie single trnsfortion fro T to T 00.
378 ONGRUENE N TRNSFORMTIONS (hpter 17) 2 squre piee of pper is divided into four tringles s shown. Show tht tringles nd re ongruent. Hene, show tht tringles nd re ongruent. Find the re of eh tringle. P 5 Q 15 R U T S 3 Whih of the figures longside is trnsltion of? Whih figure is refletion of? Whih figure is rottion of? d esrie the trnsfortion fro to. e re ll of the figures ongruent? Eplin our nswer. 4 onsider the kite longside. Use ongruene to show tht =. Hene, show tht 4X» = 4X. Show tht [X] nd [X] hve the se length. d Wht propert of kites hs een proven in? X 5 onsider the isoseles tringle longside. Eh ngle of the tringle is triseted, or divided into 3 equl prts. The ngle trisetors eet t, E, nd F s shown. Show tht F = F. Show tht 4» = 4E, nd hene = E. Show tht 4F» = 4EF. d Hene, show tht tringle EF is lso isoseles. F E