# Points A and B are at (5 ,9 ) and (8 ,6 ), respectively. Point A is rotated counterclockwise about the origin by pi  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?

Jul 26, 2018

$C = \left(- 18 , - 24\right)$

#### Explanation:

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

• " a point "(x,y)to(-x,-y)

$A \left(5 , 9\right) \to A ' \left(- 5 , - 9\right) \text{ where A' is the image of A}$

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

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

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

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

$\textcolor{w h i t e}{\underline{c}} = 2 \left(\begin{matrix}- 5 \\ - 9\end{matrix}\right) - \left(\begin{matrix}8 \\ 6\end{matrix}\right)$

$\textcolor{w h i t e}{\underline{c}} = \left(\begin{matrix}- 10 \\ - 18\end{matrix}\right) - \left(\begin{matrix}8 \\ 6\end{matrix}\right) = \left(\begin{matrix}- 18 \\ - 24\end{matrix}\right)$

$\Rightarrow C = \left(- 18 , - 24\right)$