# Points A and B are at (4 ,9 ) and (7 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 1/2 . If point A is now at point B, what are the coordinates of point C?

May 7, 2017

The point $C$ is $\left(- \frac{5}{3} , 8\right)$

#### 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 trasformation 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

$\vec{C B} = \frac{1}{2} \vec{C A '}$

$\left(\begin{matrix}7 - x \\ 2 - y\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}9 - x \\ - 4 - y\end{matrix}\right)$

So,

$7 - x = \frac{1}{2} \left(9 - x\right)$

$14 - 2 x = 9 - 5 x$

$3 x = 9 - 14 = - 5$

$x = - \frac{5}{3}$

and

$2 - y = \frac{1}{2} \left(- 4 - y\right)$

$4 - 2 y = - 4 - y$

$y = 8$

Therefore,

point $C = \left(- \frac{5}{3} , 8\right)$