# Points A and B are at (9 ,3 ) and (4 ,8 ), respectively. Point A is rotated counterclockwise about the origin by pi  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?

Jul 6, 2017

The coordinates of point $C$ are $= \left(- \frac{31}{2} , - \frac{17}{2}\right)$

#### 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

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}9 \\ 3\end{matrix}\right) = \left(\begin{matrix}- 9 \\ - 3\end{matrix}\right)$

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

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

$\left(\begin{matrix}4 - x \\ 8 - y\end{matrix}\right) = 3 \left(\begin{matrix}- 9 - x \\ - 3 - y\end{matrix}\right)$

So,

$4 - x = 3 \left(- 9 - x\right)$

$4 - x = - 27 - 3 x$

$2 x = - 31$

$x = - \frac{31}{2}$

and

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

$8 - y = - 9 - 3 y$

$2 y = - 17$

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

Therefore,

The point $C = \left(- \frac{31}{2} , - \frac{17}{2}\right)$