# Points A and B are at (1 ,8 ) and (3 ,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?

##### 1 Answer
Oct 26, 2017

$C = \left(13 , - 4\right)$

#### Explanation:

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

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(1 , 8\right) \to A ' \left(8 , - 1\right) \text{ where "A'" is the image of A}$

$\text{under a dilatation about C of factor 2}$

$\vec{C B} = 2 \vec{C A '}$

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

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

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

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = 2 \left(\begin{matrix}8 \\ - 1\end{matrix}\right) - \left(\begin{matrix}3 \\ 2\end{matrix}\right)$

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = \left(\begin{matrix}16 \\ - 2\end{matrix}\right) - \left(\begin{matrix}3 \\ 2\end{matrix}\right) = \left(\begin{matrix}13 \\ - 4\end{matrix}\right)$

$\Rightarrow C = \left(13 , - 4\right)$