# Points A and B are at (9 ,4 ) and (7 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  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?

May 28, 2018

$C = \left(\frac{5}{2} , - \frac{29}{2}\right)$

#### Explanation:

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

• " a point "(x,y)to(y,-x)

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

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

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

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

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

$\textcolor{w h i t e}{2 \underline{c}} = 3 \left(\begin{matrix}4 \\ - 9\end{matrix}\right) - \left(\begin{matrix}7 \\ 2\end{matrix}\right)$

$\textcolor{w h i t e}{2 \underline{c}} = \left(\begin{matrix}12 \\ - 27\end{matrix}\right) - \left(\begin{matrix}7 \\ 2\end{matrix}\right) = \left(\begin{matrix}5 \\ - 29\end{matrix}\right)$

$\underline{c} = \frac{1}{2} \left(\begin{matrix}5 \\ - 29\end{matrix}\right) = \left(\begin{matrix}\frac{5}{2} \\ - \frac{29}{2}\end{matrix}\right)$

$\Rightarrow C = \left(\frac{5}{2} , - \frac{29}{2}\right)$