pusat O (0,0) sejauh 120?

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Pertanyaan :


Diketahui sebuah segitiga dengan


koordinat titik A (-3,-2), B (6,-2) dan C
(6,12). Tentukanlah koordinat
bayangannya karena rotasi pada titik
pusat O (0,0) sejauh 120?

Jawaban :

\begin{pmatrix}
x\\ 
y
\end{pmatrix}
=\begin{pmatrix}
\cos\theta  & -\sin\theta \\ 
 \sin\theta &\cos\theta  
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
=\begin{pmatrix}
\cos120^{\circ}  & -\sin120^{\circ} \\ 
 \sin120^{\circ} &\cos120^{\circ}  
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
=\begin{pmatrix}
-\frac{1}{2}  & -\frac{\sqrt{3}}{2} \\ 
\frac{\sqrt{3}}{2} &-\frac{1}{2}  
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
\begin{pmatrix}
x\\ 
y
\end{pmatrix}
=\begin{pmatrix}
\frac{-x-\sqrt{3}y}{2}\\ 
\frac{\sqrt{3}x-y}{2}
\end{pmatrix}
P(x,y)\overset{R(O,120^{\circ})}{\rightarrow}P\left ( \frac{-x-\sqrt{3}y}{2},\frac{\sqrt{3}x-y}{2}\right )

A(-3,-2)\overset{R(O,120^{\circ})}{\rightarrow}A\left ( \frac{3-\sqrt{3}(-2)}{2},\frac{\sqrt{3}(-3)+2}{2}\right )=A\left ( \frac{3+2\sqrt{3}}{2},\frac{2-3\sqrt{3}}{2}\right )
B(6,-2)\overset{R(O,120^{\circ})}{\rightarrow}B\left ( \frac{-6-\sqrt{3}(-2)}{2},\frac{\sqrt{3}(6)+2}{2}\right )=B\left ( -3+\sqrt{3},1+3\sqrt{3}\right )
C(6,12)\overset{R(O,120^{\circ})}{\rightarrow}C\left ( \frac{-6-\sqrt{3}(12)}{2},\frac{\sqrt{3}(6)-12}{2}\right )=C\left ( -3-6\sqrt{3},-6+3\sqrt{3}\right )