# Points A and B are at (9 ,7 ) and (2 ,5 ), 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?

Jul 9, 2017

$C = \left(- \frac{23}{2} , 11\right)$

#### Explanation:

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

• " a point " (x,y)to(-y,x)

$\Rightarrow A \left(9 , 7\right) \to A ' \left(- 7 , 9\right) \text{where " A' " is the image of A}$

$\text{under a dilatation about C of factor 3}$

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

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

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

$\textcolor{w h i t e}{\Rightarrow 2} = 3 \left(\begin{matrix}- 7 \\ 9\end{matrix}\right) - \left(\begin{matrix}2 \\ 5\end{matrix}\right) = \left(\begin{matrix}- 21 \\ 27\end{matrix}\right) - \left(\begin{matrix}2 \\ 5\end{matrix}\right)$

$\Rightarrow \underline{c} = \frac{1}{2} \left(\begin{matrix}- 23 \\ 22\end{matrix}\right)$

$\textcolor{w h i t e}{\times \times} = \left(\begin{matrix}- \frac{23}{2} \\ 11\end{matrix}\right)$

$\text{the components of "ulc" are the coordinates of C}$

$\Rightarrow C = \left(- \frac{23}{2} , 11\right)$