# Points A and B are at (7 ,1 ) and (3 ,7 ), respectively. Point A is rotated counterclockwise about the origin by pi  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?

##### 1 Answer
Apr 20, 2018

$C = \left(- 12 , - 5\right)$

#### Explanation:

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

• " a point "(x,y)to(-x,-y)

$\Rightarrow A \left(7 , 1\right) \to A ' \left(- 7 , - 1\right) \text{ where A' is the image of A}$

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

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

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

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

$\textcolor{w h i t e}{\Rightarrow 2 \underline{c}} = 3 \left(\begin{matrix}- 7 \\ - 1\end{matrix}\right) - \left(\begin{matrix}3 \\ 7\end{matrix}\right)$

$\textcolor{w h i t e}{\Rightarrow 2 \underline{c}} = \left(\begin{matrix}- 21 \\ - 3\end{matrix}\right) - \left(\begin{matrix}3 \\ 7\end{matrix}\right) = \left(\begin{matrix}- 24 \\ - 10\end{matrix}\right)$

$\Rightarrow \underline{c} = \frac{1}{2} \left(\begin{matrix}- 24 \\ - 10\end{matrix}\right) = \left(\begin{matrix}- 12 \\ - 5\end{matrix}\right)$

$\Rightarrow C = \left(- 12 , - 5\right)$