$\begin{array}{ll}\\ 16.&\textrm{Bayangan titik A(2,4) dicerminkan }\\ &\textrm{terhadap garis}\: \: y-x=0\: \: \textrm{dilanjutkan}\\ &\textrm{ke garis}\: \: x\sqrt{3}-3y=0\: \: \textrm{adalah}\, ...\\ &\begin{array}{lll}\\ \textrm{a}.\quad \color{red}A'\left ( 2+\sqrt{3},-1+2\sqrt{3} \right )&&\\ \textrm{b}.\quad A'\left ( 2+\sqrt{3},1-2\sqrt{3} \right )&\\ \textrm{c}.\quad A'\left ( 1-\sqrt{3},-2+\sqrt{3} \right )&\\ \textrm{d}.\quad A'\left ( -2+\sqrt{3},1+2\sqrt{3} \right )\\ \textrm{e}.\quad A'\left ( 2-\sqrt{3},1-2\sqrt{3} \right ) \end{array}\\\\ &\textbf{Jawab}:\quad \textbf{a}\\ &\begin{aligned}\textrm{Dike}&\textrm{tahui bahwa}:\\ &\begin{cases} x\sqrt{3}-3y=0 & \Leftrightarrow y=\displaystyle \frac{1}{3}\sqrt{3}x\\ &\Leftrightarrow y=\tan 30^{\circ}.x\\\\ x-y=0 & \Leftrightarrow y=x \end{cases}\\\\ \begin{pmatrix} x'\\ y' \end{pmatrix}&=\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} \cos 2.30^{\circ} & \sin 2.30^{\circ} \\ \sin 2.30^{\circ} & -\cos 2.30^{\circ} \end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} \displaystyle \frac{1}{2} & \displaystyle \frac{1}{2}\sqrt{3}\\ \displaystyle \frac{1}{2}\sqrt{3} & -\displaystyle \frac{1}{2} \end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} 2\\ 4 \end{pmatrix}\\ &=\begin{pmatrix} \sqrt{3}+2\\ -1+2\sqrt{3} \end{pmatrix} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 17.&\textrm{Jika}\: \: T_{1}=\begin{pmatrix} 1 & 2\\ 1 & 1 \end{pmatrix}\: \: \textrm{dan}\: \: T_{2}=\begin{pmatrix} -2 & 5\\ -1 & 3 \end{pmatrix}\\ &\textrm{maka bayangan garis}\: \: x+y+1=0\\ &\textrm{oleh}\: \: T_{2}\circ T_{1}\: \: \textrm{adalah}\, ...\\ &\begin{array}{lll}\\ \textrm{a}.\quad \color{red}x-2y-1=0&&\\ \textrm{b}.\quad x+2y-1=0&\\ \textrm{c}.\quad x+2y+1=0&\\ \textrm{d}.\quad x-2y+1=0\\ \textrm{e}.\quad x+y-1=0 \end{array}\\\\ &\textbf{Jawab}:\quad \textbf{a}\\ &\begin{aligned}\textrm{Dike}&\textrm{tahui bahwa}:\\ \begin{pmatrix} x'\\ y' \end{pmatrix}&=T_{2}\circ T_{1}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} -2 & 5\\ -1 & 3 \end{pmatrix}\begin{pmatrix} 1 & 2\\ 1 & 1 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} -2+5 & -4+5\\ -1+3 & -2+3 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} 3 & 1\\ 2 & 1 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} 3x+y\\ 2x+y \end{pmatrix}\\ \textrm{Dipe}&\textrm{roleh}\\ &\begin{array}{lllllllll}\\ \quad x'&=3x+y\\ \quad y'&=2x+y\qquad\quad-\\\hline x'-y'&=x\\ \Leftrightarrow \quad x&=\color{red}x'-y'\qquad\color{black}....(1)\\ \color{blue}\textrm{maka}\\ \qquad y&=x'-3x\\ &=x'-3(x'-y')\\ &=\color{red}3y'-2x'\quad \color{black}....(2) \end{array}\\ \textrm{Sehin}&\textrm{gga}\\ x+y&+1=0\\ x'-&y'+3y'-2x'+1=0\\ -x'+&2y'+1=0\\ x'-&2y'-1=0\\ \textrm{maka}&\: \textrm{bayangan garisnya}\\ x-2&y-1=0 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 18.&\textrm{Garis}\: \: 2x+y+4=0\: \: \textrm{ditranslasikan}\\ &\textrm{oleh}\: \: \begin{pmatrix} -2\\ 5 \end{pmatrix}\: \: \textrm{dilanjutkan transformasi} \\ &\textrm{oleh}\: \: \begin{pmatrix} 1 & 2\\ 0 & 1 \end{pmatrix}\: \: \textrm{persamaan bayangannya}\\ &\textrm{adalah}\, ...\\ &\begin{array}{lll}\\ \textrm{a}.\quad 2x+y+3=0&&\\ \textrm{b}.\quad \color{red}2x-3y+3=0&\\ \textrm{c}.\quad 2x+3y+3=0&\\ \textrm{d}.\quad 3x+2y+3=0\\ \textrm{e}.\quad 3x-2y+3=0 \end{array}\\\\ &\textbf{Jawab}:\quad \textbf{b}\\ &\begin{aligned}\textrm{Diket}&\textrm{ahui bahwa}:\\ \begin{pmatrix} x'\\ y' \end{pmatrix}\: &=\begin{pmatrix} x\\ y \end{pmatrix}+\begin{pmatrix} -2\\ 5 \end{pmatrix}=\begin{pmatrix} x-2\\ y+5 \end{pmatrix}\\ \begin{pmatrix} x''\\ y'' \end{pmatrix}&=\begin{pmatrix} 1 & 2\\ 0 & 1 \end{pmatrix}\begin{pmatrix} x'\\ y' \end{pmatrix}\\ &=\begin{pmatrix} 1 & 2\\ 0 & 1 \end{pmatrix}\begin{pmatrix} x-2\\ y+5 \end{pmatrix}\\ &=\begin{pmatrix} x-2+2y+10\\ y+5 \end{pmatrix}\\ &=\begin{pmatrix} x+2y+8\\ y+5 \end{pmatrix}\\ \textrm{Diper}&\textrm{oleh}\\ &\begin{array}{lllllllll}\\ \quad x''&=x+2y+8\\ \: \: \: 2y''&=2y+10\qquad\quad-\\\hline x''-2y''&=x-2\\ \Leftrightarrow \quad x&=\color{red}x''-2y''+2\: \color{black}....(1)\\ \color{blue}\textrm{maka}\\ \qquad y&=\color{red}y''-5\quad \color{black}\qquad....(2) \end{array}\\ \textrm{sehin}&\textrm{gga}\\ 2x+&y+4=0\\ 2(x''&-2y''+2)+(y''-5)+4=0\\ 2x''-&3y''+3=0\\ \textrm{maka}&\: \textrm{bayangan garisnya}\\ 2x-&3y+3=0 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 19.&\textrm{Diketahui}\: \: M\: \: \textrm{adalah pencerminan terhadap}\\ &\textrm{garis}\: \: y=-x\: \: \textrm{dan}\: \: T\: \: \textrm{adalah transformasi} \\ &\textrm{yang dinyatakan oleh matriks}\: \: \begin{pmatrix} 2 & 3\\ 0 & -1 \end{pmatrix}\\ &\textrm{Koordinat bayangan titik}\: \: A(2,-8)\: \: \textrm{oleh}\\ &\textrm{transformasi}\: \: M\: \: \textrm{dilanjutkan oleh}\: \: T\: \: \textrm{adalah}\, ...\\ &\begin{array}{lll}\\ \textrm{a}.\quad (-10,2)&&\\ \textrm{b}.\quad (-2,-10)&\\ \textrm{c}.\quad \color{red}(10,2)&\\ \textrm{d}.\quad (-10,-2)\\ \textrm{e}.\quad (2,10) \end{array}\\\\ &\textbf{Jawab}:\quad \textbf{c}\\ &\begin{aligned}\textrm{Dike}&\textrm{tahui bahwa}:\\ \begin{pmatrix} x'\\ y' \end{pmatrix} &=\color{purple}T\circ M\begin{pmatrix} x\\ y \end{pmatrix}\\ &=\begin{pmatrix} 2 & 3\\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & -1\\ -1 & 0 \end{pmatrix}\begin{pmatrix} 2\\ -8 \end{pmatrix}\\ &=\begin{pmatrix} 0-3 & -2+0\\ 0+1 & 0+0 \end{pmatrix}\begin{pmatrix} 2\\ -8 \end{pmatrix}\\ &=\begin{pmatrix} -3 & -2\\ 1 & 0 \end{pmatrix}\begin{pmatrix} 2\\ -8 \end{pmatrix}\\ &=\begin{pmatrix} -6+16\\ 2+0 \end{pmatrix}\\ &=\color{red}\begin{pmatrix} 10\\ 2 \end{pmatrix} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 20.&\textrm{Jika}\: \: W\: \: \textrm{adalah transformasi oleh}\\ &\textrm{matriks}\: \: \begin{pmatrix} 1 & 0\\ 3 & 1 \end{pmatrix},\: \: \textrm{maka titik mula}\\ &\textrm{dari}\: \: W'(-2,5)\: \: \textrm{adalah}\, ...\\ &\begin{array}{lll}\\ \textrm{a}.\quad (-11,-2)&&\\ \textrm{b}.\quad (11,-2)&\\ \textrm{c}.\quad \color{red}(-2,11)&\\ \textrm{d}.\quad (2,11)\\ \textrm{e}.\quad (12,11) \end{array}\\\\ &\textbf{Jawab}:\quad \textbf{c}\\ &\begin{aligned}\textrm{Dimi}&\textrm{salkan}:\\ A&=\begin{pmatrix} -2\\ 5 \end{pmatrix},\: \: \textrm{dan}\\ W&=\begin{pmatrix} 1 & 0\\ 3 & 1 \end{pmatrix},\: \: \textrm{serta}\: \: X=\begin{pmatrix} x\\ y \end{pmatrix}\\ \textrm{mak}&\textrm{a}\\ &\begin{array}{|c|}\hline \color{red}\begin{aligned}A&=BX\\ B^{-1}A&=B^{-1}BX\\ B^{-1}A&=I.X\\ B^{-1}A&=X\\ X&=B^{-1}A \end{aligned}\\\hline \end{array}\\ \begin{pmatrix} x\\ y \end{pmatrix}&=\displaystyle \frac{1}{\begin{vmatrix} 1 &0 \\ 3 & 1 \end{vmatrix}}\begin{pmatrix} 1 & 0\\ -3 & 1 \end{pmatrix}\begin{pmatrix} -2\\ 5 \end{pmatrix}\\ &=1.\begin{pmatrix} -2+0\\ 6+5 \end{pmatrix}\\ &=\color{red}\begin{pmatrix} -2\\ 11 \end{pmatrix} \end{aligned} \end{array}$
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