# Points A and B are at (4 ,9 ) and (7 ,5 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 1/2 . If point A is now at point B, what are the coordinates of point C?

Jan 24, 2018

$C = \left(5 , 14\right)$

#### Explanation:

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

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(4 , 9\right) \to A ' \left(9 , - 4\right) \text{ where A' is the image of A}$

$\Rightarrow \vec{C B} = \textcolor{red}{\frac{1}{2}} \vec{C A '}$

$\Rightarrow \underline{b} - \underline{c} = \frac{1}{2} \left(\underline{a} ' - \underline{c}\right) = \frac{1}{2} \underline{a} ' - \frac{1}{2} \underline{c}$

$\Rightarrow - \frac{1}{2} \underline{c} = \frac{1}{2} a ' - \underline{b}$

$\textcolor{w h i t e}{\times \times \times} = \frac{1}{2} \left(\begin{matrix}9 \\ - 4\end{matrix}\right) - \left(\begin{matrix}7 \\ 5\end{matrix}\right) = \left(\begin{matrix}- \frac{5}{2} \\ - 7\end{matrix}\right)$

$\Rightarrow \underline{c} = - 2 \left(\begin{matrix}- \frac{5}{2} \\ - 7\end{matrix}\right) = \left(\begin{matrix}5 \\ 14\end{matrix}\right)$

$\Rightarrow C = \left(5 , 14\right)$