# Points A and B are at (3 ,5 ) 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 6, 2017

The coordinates of point are $C = \left(\frac{13}{2} , - 7\right)$

#### 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

$A ' = \left(\begin{matrix}0 & 1 \\ - 1 & 0\end{matrix}\right) \left(\begin{matrix}3 \\ 5\end{matrix}\right) = \left(\begin{matrix}5 \\ - 3\end{matrix}\right)$

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

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

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

So,

$2 - x = 3 \left(5 - x\right)$

$15 - 3 x = 2 - x$

$2 x = 13$

$x = \frac{13}{2}$

and

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

$- 9 - 3 y = 5 - y$

$2 y = - 14$

$y = - 7$

Therefore,

point $C = \left(\frac{13}{2} , - 7\right)$