BAB XXI. TRANSFORMASI GEOMETRI

BAB XXI. TRANSFORMASI GEOMETRI Trnsformsi digunn untu untu memindhn sutu titi tu ngun pd sutu idng. Trnsformsi geometri dlh gin dri geometri ng memhs tentng peruhn (let,entu, penjin ng didsrn dengn gmr dn mtris. Trnsformsi pd idng terdiri dri 4 mcm :. Pergesern (Trnslsi. Pencerminn (Reflesi. Perputrn (Rotsi 4. Perlin (Diltsi A. Pergesern (Trnslsi Perpindhn titi-titi pd idng dengn jr dn rh tertentu ng diwili oleh rus gris errh (vector AB tu dengn sutu psngn ilngn misl. Trnslsi T = memetn titi P(, e titi P ( +, + ng dinotsin dengn : T = : P(, P ( +, + contoh: Jdi ngn titi P(,5 oleh trnslsi T= dlh (, 8 B. Pencerminn (Reflesi Trnsformsi ng memindhn titi-titi dengn menggunn sift ngn oleh sutu cermin.. Pencerminn terhdp sumu X (dilmngn dengn M X M : P(, P (, = P (, - Persmn mtrisn : =. Pencerminn terhdp sumu Y (dilmngn dengn M Y M Y : P(, P (, = P (-, Persmn mtrisn : =. Pencerminn terhdp titi sl O(, (dilmngn dengn M O M O : P(, P (, = P (-, - Persmn mtrisn : Bngn titi P(,5 oleh trnslsi dlh = jw: T = : P(,5 P ( + (-, 5 + 4. Pencerminn terhdp gris = (dilmngn dengn M M = = : P(, P (, = P (, www.mtemti-sm.com -

. SOAL-SOAL TRANSFORMASI GOMETRI M rotsi terhdp R[, 8 ] = cos8 sin8 sin8 cos8 UAN. Bngn gris = + ng dicerminn terhdp gris = dlh: A. = + C. = - E. = - = Rotsi sudut-sudut ng lin dpt dihitung sendiri menggunn idh trigonometri tu is diliht di tel (Rngumn teori. B. = D. = + Jw: rumus dsrn : P(, P (, (. pencerminn terhdp gris = - P(, P (-, -, mtrisn pencerminn terhdp gris = : P(, P (,.( Dri ( dn ( m : = dn = ( sustitusin ( e gris = + = + = - = - Hsil pencerminnn dlh : = - jwnn dlh C UAN5. Persmn ngn urv = - oleh rotsi [, 8 ], emudin dilnjutn oleh pencerminn terhdp gris = - dlh. A. = - D. = - B. = - + E. = + + C. = + + jw: cos sin. Rotsi terhdp R [, ] = sin cos Bngn oleh oleh rotsi [, 8 ], emudin dilnjutn oleh pencerminn terhdp gris = - dlh : = = = = (, ; = sustitusin pd urv = - = - - = - jwnn dlh D EBTANAS99. Persmn ngn dri lingrn + +4-6-= oleh trnsformsi ng eritn dengn mtris dlh. A. + - 6-4- = B. + - 6 + 4- = C. + + 6-4- = D. + - 4 + 6- = E. + + 4-6+ = www.mtemti-sm.com -

Jw: = = = dn = - - = sustitusin pd persmn lingrn + +4-6-= menjdi : EBTANAS99 5. Ditentun mtris trnsformsi T = dn T =. Hsil trnsformsi titi (,- terhdp T dilnjutn T dlh. A. (-4, B. (-,4 C. (,4 D. (4, E. (,-4 jw: Trnsformsi T dilnjutn oleh T = T o T (- + ( - 4-6 - = + - 4-6 - = T o T = M M =. = + 6-4 = Jwnn dlh A EBTANAS995 4. T dn T dlh trnsformsi ng msing-msing ersesuin dengn dn. Ditentun T = T o T, m trnsformsi T ersesuin dengn mtris 7 A. 4 4 5 B. 7 C. D. 5 4 E. 7 Jw: M = mtris trnsformsi T = M = mtris trnsformsi T = T = T o T = M M Hsil trnsformsi titi (,- terhdp T dilnjutn T 4 dlh. = ( -4, Jwnn dlh A UN5 6. Persmn ngn gris = -6 + ren trnsformsi oleh mtris emudin dilnjutn dengn mtris dlh A. + + = D. + + 9 = B. + = E. + 9 = C. 8 9 + = Jw: Mtris T = M = MtrisT = M =. (. +. (. +.(. = (. +. (.+.( Jwnn dlh E 4 = 7 Trnsformsi T dilnjutn T dlh = T o T =M M M M = 4 = 4 5 www.mtemti-sm.com - 4. = 4 5

Ingt mtris : Ji A.B = C m. A = C. B. B = A. C A.B = C C = A.B 4 C = ; A = 4 5 B = A. C ; B = UAN4 7. Bngn titi A (4, oleh pencerminn terhdp gris = dilnjutn pencerminn terhdp gris = 5 dlh titi. A. A (8,5 C. A (8, E. A (, B. A (, D. A (4,5 Jw:. Cr (dengn rumus Pencerminn terhdp gris = h P(, P (, = P (h, A = ( 4.4 5 4 4 A(4, = A ((-4, 5 4 = ( 4.4 4 5 4 5 4 = = 6 6 6 4 4 6 6 5 4 = 6 6 4 6 6 A (, = 5 A (.5, A (,. Cr ( dengn gmr = 6 5 + 6 4 = - 6 4-6 sustitusin pd persmn gris = -6 + - 6 4-6 = - 5-4 + titi A (4, dicerminn terhdp gris = didpt A (, emudin dicerminn lgi terhdp gris =5 didpt A (, - 6 4 + 5-6 + 4 - = Jwnn dlh B 4 + + 4 + - = 6 6 6 + - = 6 6 6 6 + - 8 = : + - 9 = + 9 = Jwnn dlh E UAN4 8. T dlh trnsformsi ng ersesuin dengn mtris 5 dn T dlh trnsformsi ng ersesuin dengn mtris. 4 Bngn A(m,n oleh trnsformsi T o T dlh (-9,7. Nili m+n sm dengn A. 4 B.5 C.6 D.7 E.8 www.mtemti-sm.com -

Jw: 9 m = M M 7 n 9 5 m = 7 4 n 9 m = 7 5 n - m - n = -9-5m + n = 7 - m - n = -9 5-5m 5n = -45-5m + n = 7-5m +n = 7 - -6n = -5 n = - m n = - 9 -m = n 9 m = 9 n = 9. = 9 6 = Sehingg m+ n = + = 5 Jwnn dlh B UAN 9. Bngn ABC dengn A(,, B(6,, C(5, ren reflesi terhdp sumu dilnjutn rotsi (,9 dlh A. A (-,-, B (,6 dn C (-,-5 B. A (-,-, B (,-6 dn C (-,-5 C. A (,-, B (-,6 dn C (-,5 D. A (-,-, B (-,-6 dn C (-,-5 E. A (-,, B (-,-6 dn C (-,-5 jw: Pencerminn/reflesi terhdp sumu Y P(, P (-,. Rotsi (, 9 : cos sin cos9 sin 9 sin cos sin 9 cos9 = = - ; = Rumus lngsung: P(, P (-, s: rotsi (,9 : P(, P (-, P (-, - cttn: dri P (-, dirotsi (,9 menjdi P (-, - didpt dri rumus rotsi (,9 P(, P (-, sehingg : (-, (-, A(, A (-, A " (-,- B(6, B (-6, B " (-,-6 C(5, C (-5, C " (-,-5. Cr (lngsung reflesi terhdp sumu Y dilnjutn rotsi (,9 : s: rotsi (,9 : (-, (-, P(, P (-, P (-, - cttn: dri P (-, dirotsi (,9 menjdi P (-, - didpt dri rumus rotsi (,9 P(, P (-, P(, P (-, - A(, A " (-,- B(6, B " (-,-6 C(5, C " (-,-5 Jwnn dlh D UAN. Persmn pet urv = - + ren pencerminn terhdp sumu dilnjutn diltsi dengn pust O dn fctor sl dlh A. + - 9 + 8 = B. - + 9-8 = C. - + 9 + 8 = D. + + 9 + 8 = E. + + 9-8 = www.mtemti-sm.com - 4

Jw: pencerminn terhdp sumu : P(, P (, - Diltsi terhdp titi pust O(, dengn fctor sl : [O, ] : P(, P (, [O,] : P(, P (, pencerminn terhdp sumu dilnjutn diltsi dengn pust O dn fctor sl : P(, P (, - P (, - " = = " " = - = - " Sehingg : P(, P " (-, P(-,, Q(,, R (,-, S(-,- P(-, P " (-6,- Q(, Q " (-6,9 R (,- R " (,9 S(-,- S " (,- But sets gmrn: Q " (-6,9 Q " (-6,9 9 Sustitusi pd persmn = - + menjdi: - " = ( " -. " + - " = 9 " - " + 9 - " = " - 9 " + 8 " + " - 9 " + 8 = + - 9 + 8 = jwnn dlh A EBTANAS. Lus ngn persegipnjng PQRS dengn P(-,, Q(,, R (,-, S(-,- ren diltsi [,] dilnjutn rotsi pust O ersudut π dlh A. 6 B. 48 C.7 D. 96 E. 8 jw: diltsi [,] : [O,] : P(, P (, Rotsi pust O ersudut π { R [, π ] }: P(, P (-, [,] (-, P(, P (, P " (-, -6 P " (-6,- - S " (,- (6+ stun Sehingg lus trnsformsin dlh : Pnjng (p ler (l = 9 = 8 stun lus jwnn dlh E (9+ stun lus EBTANAS. Segitig ABC dengn A(,, B(6,, C(6,4 ditrnsformsin dengn mtris trnsformsi. Lus ngun hsil trnsformsi segitig ABC dlh. A. 56 stun lus C. 8 stun lus E. 8 stun lus B. 6 stun lus D. 4 stun lus Jw: misln T = m Lus ngn/trnsformsi ABC = det T lus ABC det T = d c = - = lus ABC : www.mtemti-sm.com - 5

ut sets gmr: 4 C(6,4 A(, B(6, 6 7 Lus ABC = ls tinggi ; = AB BC =. 4 = 6 Lus ngn/trnsformsi ABC = det T lus ABC = 6 = 8 stun lus Jwnn dlh E www.mtemti-sm.com - 6

Persmn mtrisn : = 5. Pencerminn terhdp gris = - (dilmngn dengn M M = = : P(, P (, = P (-, - Persmn mtrisn : = 6. Pencerminn terhdp gris = h (dilmngn dengn M = h M = : P(, P (, = P (h, h 7. Pencerminn terhdp gris = (dilmngn dengn M M = = : P(, P (, = P (, - 8. Pencerminn terhdp titi (, (dilmngn dengn M (,.( +.5 =.( +.5 5 = 5 Jdi titi ngn A dlh A. Bngn gris = - ng dicerminn terhdp gris = - dlh.. Jw: = = = - = - = - = - sustitusin e persmn gris = menjdi: - = (- = - Jdi ngnn dlh = - Contoh: M (, : P(, P (, = P ( -, -. Titi A(-, 5 dicerminn terhdp gris =, ordint titi ngn A dlh Jw: C. Perputrn (Rotsi Trnsformsi ng memindhn titi-titi dengn memutr titi-titi terseut sejuh terhdp sutu titi pust rotsi. Sutu rotsi dengn pust P dn sudut rotsi dinotsin dengn R (P,. = = 5. Rotsi terhdp titi pust O(, (dilmngn dengn R(O, Ji titi P(, diputr seesr elwnn rh jm Terhdp titi pust O(,, m diperoleh ngn P (,. R(O, : P(, P (, = P ( cos - sin, sin + cos www.mtemti-sm.com -

www.mtemti-sm.com - Persmn mtrin: = cos sin sin cos Untu = 9, -9, 8, 7, -7 dengn memsun nili terseut didpt tle s: Rotsi Bngn Mtris R(O, 9 (-, R(O, -9 (, - R(O, 8 (-, - R(O, 7 (, - R(O, -7 (-,. Rotsi terhdp titi pust P(, (dilmngn dengn R(O, Ji sutu titi P (, diputr sejuh erlwnn dengn rh jm terhdp titi pust A(, m ngnn dlh P (, dengn - = ( cos - (- sin - = ( sin + (- cos Persmn mtrin: = cos sin sin cos + Contoh sol:. Titi B(, dirotsin terhdp titi (,. Tentun Bngn titi B pil titi B dirotsin. sejuh 9 erlwnn rh dengn jrum jm. sejuh 9 serh jrum jm Jw:. = = =. = = = D. Perlin tu Diltsi Trnsformsi ng menguh jr titi-titi dengn fctor pengli tertentu terhdp sutu titi tertentu. Perlin tu diltsi ini ditentun oleh fctor sl ( dn pust diltsi.. Diltsi terhdp titi pust O(, Pemetnn: [O, ] : P(, P (, persmn mtrisn : =. Diltsi terhdp titi pust A(, Titi P(, diltsi terhdp titi pust A (, dengn fctor sl, didpt ngn P (, dengn:

www.mtemti-sm.com - 4 - = ( - dn - = ( - Persmn mtrisn : = + Contoh:. Bngn titi B(, diltsi terhdp titi pust O(, dengn fctor sl dlh: Jw: = =, = ; = msun e dlm pers mtris: = didpt : = dn = 6 Jdi ngn titi B(, diltsi terhdp titi pust O (, dengn fctor sl dlh B (,6. Bngn titi B(-, diltsi terhdp titi pust A(, dengn fctor sl - dlh: jw: = + = - ; = - ; = ; = ; ; msun e dlm persmn mtris: = + = + = + = 7 7 Jdi ngn titi B(-, diltsi terhdp titi pust A(, dengn sl - dlh B ( 7, 7 E. Trnsformsi oleh sutu Mtris. Sutu titi A (, ditrnsformsin oleh mtris d c menjdi A (,. Huungn di ts dpt ditulisn dlm persmn mtris: = d c Contoh: Hsil trnsformsi mtris 5 4 terhdp titi B(, - dlh jw: = d c = 5 4

8 = 9 Jdi B dlh (-8, -9 F. Kompisisi Trnsformsi Gungn dri eerp trnsformsi diseut dengn omposisi trnsformsi. Trnsformsi T dilnjutn dengn trnsformsi T terhdp sutu titi P (, : Dlm entu gn urutn trnsformsi dpt diperlihtin s: T T P(, P (, P (, Pengerjn trnsformsi ini dpt ditulis dengn: T o T P(, T o T. Komposisi du trnslsi Ji trnslsi T P (, c dn T, d omposisi trnslsi T dilnjutn dengn T dpt diwili oleh trnslsi tunggl ng ditentun oleh: c + c T o T = + = d + d sift-sift omposisi trnslsi. Untu du trnslsi erurutn erlu T o T = T o T (omuttif. Untu tig trnslsi erurutn erlu (T o T o T = T o ( T o T (sositif contoh: Titi B(,4 ditrnslsin oleh T emudin 4 dilnjutn dengn T, ngn titi B dlh jw: c + c T = T o T = + = d + d + 4 = = + 4 6 4 6 = + = 4 6 jdi ngnn dlh (6,. Komposisi Reflesi. Komposisi du reflesi terhdp sumu-sumu sejjr. Sejjr terhdp sumu Ji titi P (, dlh hsil pencerminn terhdp gris = dn titi P (, dlh hsil pencerminn titi P (, terhdp gris =. (liht gmr P (, = P (, P (, = P (, P (, P (, = P P P = (, (- (, ( + (, d + d = jr ntr du sumu ng sejjr www.mtemti-sm.com - 5

Jdi ji trnsformsi pencerminn terhdp gris = diseut dengn M = dn trnsformsi pencerminn terhdp gris = diseut dengn M =, m P (, M = o M = P (, d + ; d =. Komposisi du reflesi terhdp sumu-sumu sling teg lurus Ji titi P (, dlh hsil pencerminn titi P (, terhdp gris = dn titi P (, dlh hsil pencerminn titi P (, tehdp gris =.. Sejjr terhdp sumu Ji titi P (, dlh hsil pencerminn terhdp gris = dn titi P (, dlh hsil pencerminn titi P (, terhdp gris = (liht gmr P (, P (, P (, M: = P (, P ( (-, = = = P (, P ( (-, P (-, = P P P P ( (-, ( (- +, ( ((- +, ( d +, d = jr ntr du sumu ng sejjr Jdi ji trnsformsi pencerminn terhdp gris = diseut dengn M = dn Trnsformsi pencerminn terhdp gris = diseut dengn M =, m M = o M = (, P (d +, ; d = P (-, = Jdi M = o M = P (, P (-, - P (-, - Pencerminn terhdp du sumu ng sling teg lurus euivlen dengn rotsi pust perpotongn du sumu dn sudut putr 8, ditulis s: M = o M = = R((,, 8 c. Komposisi du reflesi terhdp sumu-sumu sling erpotongn Pencerminn terhdp du sumu ng sling erpotongn n menghsiln rotsi ng ersift:. Titi potong edu sumu pencerminn dlh pust perputrn. Besr sudut putr dlh du li sudut ntr edu sumu pencerminn. Arh perputrn sm dengn rh dri sumu pertm e sumu edu. www.mtemti-sm.com - 6

Pemetnn dpt ditulis s: M o M = R(T, T = titi potong edu sumu = sudut ntr edu sumu. Komposisi Rotsi Du rotsi erurutn ng sepust eivlen dengn rotsi sejuh jumlh edu sudut rotsin terhdp pust ng sm. Ji R = R (, dn R = R(, β m: R o R = R(, ( + β Lus Bngun A = det T Lus ngun A det T dinmn fctor peresrn lus, merupn nili mutl determinn mtris T. det T = d c Contoh sol: Diethui segitig ABC dengn oordint A(,, B(,5, C(6,. Berp lus ngn segitig ABC oleh trnsformsi ng ersesuin dengn mtris? Jw: Komposisi Trnsformsi dengn Mtris Ji T dlh trnsformsi ng ersesuin dengn mtris M = dn T dlh trnsformsi c d c d ng ersesuin dengn mtris M = m e f omposisi trnsformsi :. T o T dlh perlin mtris M. M c d M. M = e f c d diethui ABC : Als = AC = 5 ; tinggi = AB=4 Lus ABC = ls tinggi = AC AB =. 5. 4 = stun lus. T o T dlh perlin mtris M. M M. M = c c d e Lus derh ngun hsil trnsformsi Ji mtris trnsformsi T = c d mentrnsformsin ngun A menjdi ngun A, m : d f ABC ditrnsformsin ng ersesuin dengn mtris. Misl mtris ini dlh T, m: det T =. (- = + 6 = 8 Lus ngn ABC = det T Lus ABC = 8 = 8 stun lus www.mtemti-sm.com - 7

Tel mcm-mcm Trnsformsi dn mtrisn : No Trnsformsi Notsi Mtris P( Trnslsi, P ( +, + Pencerminn terhdp sumu X P(, P (, - (Reflesi Pencerminn terhdp sumu Y P(, P (-, (Reflesi 4 Pencerminn terhdp titi sl P(, P (-, - (, 5 Pencerminn terhdp gris = P(, P (, 6 Pencerminn terhdp gris = - P(, P (-, - 7 Pencerminn terhdp gris = h P(, P (h, 8 Pencerminn terhdp gris = P(, P (, - 9 Pencerminn terhdp titi (, P(, P ( -, - Rotsi terhdp titi pust O(, R(O, erlwnn rh jm Rotsi terhdp titi pust P(, R(O, erlwnn dengn rh jm P(, P ( cos - sin, sin + cos - = ( cos - (- sin - = ( sin + (- cos Diltsi terhdp titi pust O(, [O, ] : P(, P (, Diltsi terhdp titi pust A(, - = ( - - = ( - cos sin sin cos cos sin + sin cos + * T o T Trnsformsi T dilnjutn oleh T Ji M dn M dlh mtris trnsformsi T dn T m T o T dlh M M www.mtemti-sm.com - 8