# Points A and B are at (2 ,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 2 . If point A is now at point B, what are the coordinates of point C?

Jun 24, 2017

The coordinates of point $C$ are $\left(1 , - 6\right)$

#### Explanation:

The matrix of a rotation counterclockwise by $\frac{3}{2} \pi$ about the origin is

$\left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right)$

Therefore, the transformation of point $A$ is

$A ' = \left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right) \left(\begin{matrix}2 \\ 4\end{matrix}\right) = \left(\begin{matrix}4 \\ - 2\end{matrix}\right)$

Let point $C$ be $\left(x , y\right)$, then

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

$\left(\begin{matrix}7 - x \\ 2 - y\end{matrix}\right) = 2 \left(\begin{matrix}4 - x \\ - 2 - y\end{matrix}\right)$

So,

$7 - x = 2 \left(4 - x\right)$

$7 - x = 8 - 2 x$

$x = 1$

and

$2 - y = 2 \left(- 2 - y\right)$

$2 - y = - 4 - 2 y$

$y = - 4 - 2$

$y = - 6$

Therefore,

point $C = \left(1 , - 6\right)$