Contoh Soal 4 Transformasi Geometri

$\begin{array}{l}\\ 16.& \text{Bayangan titik A(2,4) dicerminkan }\\ & \text{terhadap garis}y-x=0\text{dilanjutkan}\\ & \text{ke garis}x\sqrt{3}-3y=0\text{adalah}...\\ & \begin{array}{l}\\ \text{a}.A^{\prime }(2+\sqrt{3},-1+2\sqrt{3})& & \\ \text{b}.A^{\prime }(2+\sqrt{3},1-2\sqrt{3})& \\ \text{c}.A^{\prime }(1-\sqrt{3},-2+\sqrt{3})& \\ \text{d}.A^{\prime }(-2+\sqrt{3},1+2\sqrt{3})\\ \text{e}.A^{\prime }(2-\sqrt{3},1-2\sqrt{3})\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}:\\ & \{\begin{array}{ll}x\sqrt{3}-3y=0& \iff y={\displaystyle \frac{1}{3}\sqrt{3}x}\\ & \iff y=\tan {30}^{\circ }.x\\ \\ x-y=0& \iff y=x\end{array}\\ \\ \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =\left(\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}\cos {2.30}^{\circ }& \sin {2.30}^{\circ }\\ \sin {2.30}^{\circ }& -\cos {2.30}^{\circ }\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}\sqrt{3}}\\ {\displaystyle \frac{1}{2}\sqrt{3}}& -{\displaystyle \frac{1}{2}}\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}2\\ 4\end{array}\right)\\ & =\left(\begin{array}{c}\sqrt{3}+2\\ -1+2\sqrt{3}\end{array}\right)\end{array}\end{array}$.

$\begin{array}{l}\\ 17.& \text{Jika}{T}_1=\left(\begin{array}{cc}1& 2\\ 1& 1\end{array}\right)\text{dan}{T}_2=\left(\begin{array}{cc}-2& 5\\ -1& 3\end{array}\right)\\ & \text{maka bayangan garis}x+y+1=0\\ & \text{oleh}{T}_2\circ T_1\text{adalah}...\\ & \begin{array}{l}\\ \text{a}.x-2y-1=0& & \\ \text{b}.x+2y-1=0& \\ \text{c}.x+2y+1=0& \\ \text{d}.x-2y+1=0\\ \text{e}.x+y-1=0\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}:\\ \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =T_2\circ T_1\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}-2& 5\\ -1& 3\end{array}\right)\left(\begin{array}{cc}1& 2\\ 1& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}-2+5& -4+5\\ -1+3& -2+3\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}3& 1\\ 2& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{c}3x+y\\ 2x+y\end{array}\right)\\ \text{Dipe}& \text{roleh}\\ & \begin{array}{l}\\ x^{\prime }& =3x+y\\ y^{\prime }& =2x+y-\\ x^{\prime }-y^{\prime }& =x\\ \iff x& =x^{\prime }-y^{\prime }....(1)\\ \text{maka}\\ y& =x^{\prime }-3x\\ & =x^{\prime }-3(x^{\prime }-y^{\prime })\\ & =3y^{\prime }-2x^{\prime }....(2)\end{array}\\ \text{Sehin}& \text{gga}\\ x+y& +1=0\\ x^{\prime }-& y^{\prime }+3y^{\prime }-2x^{\prime }+1=0\\ -x^{\prime }+& 2y^{\prime }+1=0\\ x^{\prime }-& 2y^{\prime }-1=0\\ \text{maka}& \text{bayangan garisnya}\\ x-2& y-1=0\end{array}\end{array}$.

$\begin{array}{l}\\ 18.& \text{Garis}2x+y+4=0\text{ditranslasikan}\\ & \text{oleh}\left(\begin{array}{c}-2\\ 5\end{array}\right)\text{dilanjutkan transformasi}\\ & \text{oleh}\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\text{persamaan bayangannya}\\ & \text{adalah}...\\ & \begin{array}{l}\\ \text{a}.2x+y+3=0& & \\ \text{b}.2x-3y+3=0& \\ \text{c}.2x+3y+3=0& \\ \text{d}.3x+2y+3=0\\ \text{e}.3x-2y+3=0\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{b}}\\ & \begin{array}{rl}\text{Diket}& \text{ahui bahwa}:\\ \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}-2\\ 5\end{array}\right)=\left(\begin{array}{c}x-2\\ y+5\end{array}\right)\\ \left(\begin{array}{c}{x}^{″}\\ y^{″}\end{array}\right)& =\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)\\ & =\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\left(\begin{array}{c}x-2\\ y+5\end{array}\right)\\ & =\left(\begin{array}{c}x-2+2y+10\\ y+5\end{array}\right)\\ & =\left(\begin{array}{c}x+2y+8\\ y+5\end{array}\right)\\ \text{Diper}& \text{oleh}\\ & \begin{array}{l}\\ x^{″}& =x+2y+8\\ 2y^{″}& =2y+10-\\ x^{″}-2y^{″}& =x-2\\ \iff x& =x^{″}-2y^{″}+2....(1)\\ \text{maka}\\ y& =y^{″}-5....(2)\end{array}\\ \text{sehin}& \text{gga}\\ 2x+& y+4=0\\ 2(x^{″}& -2y^{″}+2)+(y^{″}-5)+4=0\\ 2x^{″}-& 3y^{″}+3=0\\ \text{maka}& \text{bayangan garisnya}\\ 2x-& 3y+3=0\end{array}\end{array}$.

$\begin{array}{l}\\ 19.& \text{Diketahui}M\text{adalah pencerminan terhadap}\\ & \text{garis}y=-x\text{dan}T\text{adalah transformasi}\\ & \text{yang dinyatakan oleh matriks}\left(\begin{array}{cc}2& 3\\ 0& -1\end{array}\right)\\ & \text{Koordinat bayangan titik}A(2,-8)\text{oleh}\\ & \text{transformasi}M\text{dilanjutkan oleh}T\text{adalah}...\\ & \begin{array}{l}\\ \text{a}.(-10,2)& & \\ \text{b}.(-2,-10)& \\ \text{c}.(10,2)& \\ \text{d}.(-10,-2)\\ \text{e}.(2,10)\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Dike}& \text{tahui bahwa}:\\ \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =T\circ M\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =\left(\begin{array}{cc}2& 3\\ 0& -1\end{array}\right)\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)\left(\begin{array}{c}2\\ -8\end{array}\right)\\ & =\left(\begin{array}{cc}0-3& -2+0\\ 0+1& 0+0\end{array}\right)\left(\begin{array}{c}2\\ -8\end{array}\right)\\ & =\left(\begin{array}{cc}-3& -2\\ 1& 0\end{array}\right)\left(\begin{array}{c}2\\ -8\end{array}\right)\\ & =\left(\begin{array}{c}-6+16\\ 2+0\end{array}\right)\\ & =\left(\begin{array}{c}10\\ 2\end{array}\right)\end{array}\end{array}$.

$\begin{array}{l}\\ 20.& \text{Jika}W\text{adalah transformasi oleh}\\ & \text{matriks}\left(\begin{array}{cc}1& 0\\ 3& 1\end{array}\right),\text{maka titik mula}\\ & \text{dari}{W}^{\prime }(-2,5)\text{adalah}...\\ & \begin{array}{l}\\ \text{a}.(-11,-2)& & \\ \text{b}.(11,-2)& \\ \text{c}.(-2,11)& \\ \text{d}.(2,11)\\ \text{e}.(12,11)\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Dimi}& \text{salkan}:\\ A& =\left(\begin{array}{c}-2\\ 5\end{array}\right),\text{dan}\\ W& =\left(\begin{array}{cc}1& 0\\ 3& 1\end{array}\right),\text{serta}X=\left(\begin{array}{c}x\\ y\end{array}\right)\\ \text{mak}& \text{a}\\ & \begin{array}{|c|}\hline \begin{array}{rl}A& =BX\\ B^{-1}A& =B^{-1}BX\\ B^{-1}A& =I.X\\ B^{-1}A& =X\\ X& =B^{-1}A\end{array}\\ \hline\end{array}\\ \left(\begin{array}{c}x\\ y\end{array}\right)& ={\displaystyle \frac{1}{\left|\begin{array}{cc}1& 0\\ 3& 1\end{array}\right|}\left(\begin{array}{cc}1& 0\\ -3& 1\end{array}\right)\left(\begin{array}{c}-2\\ 5\end{array}\right)}\\ & =1.\left(\begin{array}{c}-2+0\\ 6+5\end{array}\right)\\ & =\left(\begin{array}{c}-2\\ 11\end{array}\right)\end{array}\end{array}$