# Points A and B are at (5 ,9 ) and (8 ,6 ), respectively. Point A is rotated counterclockwise about the origin by pi  and dilated about point C by a factor of 4 . If point A is now at point B, what are the coordinates of point C?

Jul 5, 2017

The point $C = \left(- \frac{28}{3} , - 14\right)$

#### 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

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}5 \\ 9\end{matrix}\right) = \left(\begin{matrix}- 5 \\ - 9\end{matrix}\right)$

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

$\vec{C B} = 4 \vec{C A '}$

$\left(\begin{matrix}8 - x \\ 6 - y\end{matrix}\right) = 4 \left(\begin{matrix}- 5 - x \\ - 9 - y\end{matrix}\right)$

So,

$8 - x = 4 \left(- 5 - x\right)$

$8 - x = - 20 - 4 x$

$3 x = - 28$

$x = - \frac{28}{3}$

and

$6 - y = 4 \left(- 9 - y\right)$

$6 - y = - 36 - 4 y$

$3 y = - 42$

$y = - 14$

Therefore,

The point $C = \left(- \frac{28}{3} , - 14\right)$