# Points A and B are at (5 ,8 ) and (8 ,1 ), respectively. Point A is rotated counterclockwise about the origin by pi  and dilated about point C by a factor of 5 . If point A is now at point B, what are the coordinates of point C?

Jul 3, 2017

The point $C$ is $= \left(- \frac{33}{4} , - \frac{41}{4}\right)$

#### 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

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}5 \\ 8\end{matrix}\right) = \left(\begin{matrix}- 5 \\ - 8\end{matrix}\right)$

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

$\vec{C B} = 5 \vec{C A '}$

$\left(\begin{matrix}8 - x \\ 1 - y\end{matrix}\right) = 5 \left(\begin{matrix}- 5 - x \\ - 8 - y\end{matrix}\right)$

So,

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

$8 - x = - 25 - 5 x$

$4 x = - 33$

$x = - \frac{33}{4}$

and

$1 - y = 5 \left(- 8 - y\right)$

$1 - y = - 40 - 5 y$

$4 y = - 40 - 1$

$y = - \frac{41}{4}$

Therefore,

point $C = \left(- \frac{33}{4} , - \frac{41}{4}\right)$