# Points A and B are at (8 ,3 ) and (5 ,7 ), respectively. Point A is rotated counterclockwise about the origin by pi/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 23, 2017

The point $C = \left(- 7 , \frac{17}{2}\right)$

#### Explanation:

The matrix of a rotation counterclockwise by $\frac{1}{2} \pi$ about the origin is

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

Therefore, the transformation of point $A$ is

$A ' = \left(\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right) \left(\begin{matrix}8 \\ 3\end{matrix}\right) = \left(\begin{matrix}- 3 \\ 8\end{matrix}\right)$

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

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

$\left(\begin{matrix}5 - x \\ 7 - y\end{matrix}\right) = 3 \left(\begin{matrix}- 3 - x \\ 8 - y\end{matrix}\right)$

So,

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

$5 - x = - 9 - 3 x$

$2 x = - 14$

$x = - 7$

and

$7 - y = 3 \left(8 - y\right)$

$7 - y = 24 - 3 y$

$2 y = 24 - 7$

$y = \frac{17}{2}$

Therefore,

point $C = \left(- 7 , \frac{17}{2}\right)$