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

Jun 2, 2017

The coordinates of the point $C = \left(2 , - 16\right)$

#### 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 trasformation of point $A$ is

$A ' = \left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right) \left(\begin{matrix}9 \\ 2\end{matrix}\right) = \left(\begin{matrix}2 \\ - 9\end{matrix}\right)$

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

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

$\left(\begin{matrix}2 - x \\ 5 - y\end{matrix}\right) = 3 \left(\begin{matrix}2 - x \\ - 9 - y\end{matrix}\right)$

So,

$2 - x = 3 \left(2 - x\right)$

$6 - 3 x = 2 - x$

$2 x = 4$

$x = 2$

and

$5 - y = 3 \left(- 9 - y\right)$

$- 27 - 3 y = 5 - y$

$2 y = - 32$

$y = - 16$

Therefore,

point $C = \left(2 , - 16\right)$