# Points A and B are at (4 ,3 ) and (5 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 2 . If point A is now at point B, what are the coordinates of point C?

Feb 28, 2018

$C = \left(1 , - 10\right)$

#### Explanation:

$\text{under a counterclockwise rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(4 , 3\right) \to A ' \left(3 , - 4\right) \text{ where A' is the image of A}$

$\Rightarrow \vec{C B} = \textcolor{red}{2} \vec{C A '}$

$\Rightarrow \underline{b} - \underline{c} = 2 \left(\underline{a} ' - \underline{c}\right)$

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

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

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = 2 \left(\begin{matrix}3 \\ - 4\end{matrix}\right) - \left(\begin{matrix}5 \\ 2\end{matrix}\right)$

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = \left(\begin{matrix}6 \\ - 8\end{matrix}\right) - \left(\begin{matrix}5 \\ 2\end{matrix}\right) = \left(\begin{matrix}1 \\ - 10\end{matrix}\right)$

$\Rightarrow C = \left(1 , - 10\right)$