# Points A and B are at (4 ,9 ) and (6 ,8 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 4 . If point A is now at point B, what are the coordinates of point C?

Jun 15, 2017

The point $C = \left(10 , - 8\right)$

#### 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

$A ' = \left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right) \left(\begin{matrix}4 \\ 9\end{matrix}\right) = \left(\begin{matrix}9 \\ - 4\end{matrix}\right)$

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

$\vec{C B} = 4 \vec{C A '}$

$\left(\begin{matrix}6 - x \\ 8 - y\end{matrix}\right) = 4 \left(\begin{matrix}9 - x \\ - 4 - y\end{matrix}\right)$

So,

$6 - x = 3 \left(9 - x\right)$

$6 - x = 36 - 4 x$

$3 x = 30$

$x = 10$

and

$8 - y = 4 \left(- 4 - y\right)$

$8 - y = - 16 - 4 y$

$3 y = - 24$

$y = - 8$

Therefore,

point $C = \left(10 , - 8\right)$