# Points A and B are at (9 ,9 ) and (7 ,6 ), respectively. Point A is rotated counterclockwise about the origin by pi  and dilated about point C by a factor of 2 . If point A is now at point B, what are the coordinates of point C?

Jun 23, 2017

The point $C = \left(- 25 , - 24\right)$

#### Explanation:

The matrix of a rotation counterclockwise by $\pi$ about the origin is

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

Therefore, the transformation of point $A$ is

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

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

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

$\left(\begin{matrix}7 - x \\ 6 - y\end{matrix}\right) = 2 \left(\begin{matrix}- 9 - x \\ - 9 - y\end{matrix}\right)$

So,

$7 - x = 2 \left(- 9 - x\right)$

$7 - x = - 18 - 2 x$

$x = - 25$

$x = - 25$

and

$6 - y = 2 \left(- 9 - y\right)$

$6 - y = - 18 - 2 y$

$y = - 18 - 6$

$y = - 24$

Therefore,

The point $C = \left(- 25 , - 24\right)$