# Points A and B are at (2 ,9 ) and (3 ,2 ), respectively. Point A is rotated counterclockwise about the origin by pi/2  and dilated about point C by a factor of 3 . If point A is now at point B, what are the coordinates of point C?

Jun 5, 2017

$C = \left(- 15 , 2\right)$

#### Explanation:

$\text{Under a counterclockwise rotation about the origin of } \frac{\pi}{2}$

• " a point " (x,y)to(-y,x)

$\Rightarrow A \left(2 , 9\right) \to A ' \left(- 9 , 2\right) \text{ where " A' " is the image of A}$

$\text{Under a dilatation about " C" of factor 3}$

$\vec{C B} = \textcolor{red}{3} \vec{C A '}$

$\Rightarrow \underline{b} - \underline{c} = \textcolor{red}{3} \left(\underline{a} ' - \underline{c}\right)$

$\Rightarrow \underline{b} - \underline{c} = 3 \underline{a} ' - 3 \underline{c}$

$\Rightarrow 2 \underline{c} = 3 \underline{a} ' - \underline{b}$

$\textcolor{w h i t e}{\Rightarrow 2 c x} = 3 \left(\begin{matrix}- 9 \\ 2\end{matrix}\right) - \left(\begin{matrix}3 \\ 2\end{matrix}\right) = \left(\begin{matrix}- 30 \\ 4\end{matrix}\right)$

$\Rightarrow \underline{c} = \frac{1}{2} \left(\begin{matrix}- 30 \\ 4\end{matrix}\right) = \left(\begin{matrix}- 15 \\ 2\end{matrix}\right)$

$\Rightarrow C = \left(- 15 , 2\right)$