# Points A and B are at (4 ,3 ) and (1 ,4 ), respectively. Point A is rotated counterclockwise about the origin by pi/2  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?

Jun 16, 2017

$C = \left(- 7 , 4\right)$

#### Explanation:

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

• " a point " (x,y)to(-y,x)

$\Rightarrow A \left(4 , 3\right) \to A ' \left(- 3 , 4\right) \text{ where A' is the image of A}$

$\text{under a dilatation about C of factor 2}$

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

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

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

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

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = 2 \left(\begin{matrix}- 3 \\ 4\end{matrix}\right) - \left(\begin{matrix}1 \\ 4\end{matrix}\right)$

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = \left(\begin{matrix}- 6 \\ 8\end{matrix}\right) - \left(\begin{matrix}1 \\ 4\end{matrix}\right) = \left(\begin{matrix}- 7 \\ 4\end{matrix}\right)$

$\Rightarrow C = \left(- 7 , 4\right)$