# Points A and B are at (4 ,6 ) and (8 ,3 ), respectively. Point A is rotated counterclockwise about the origin by pi  and dilated about point C by a factor of 1/2 . If point A is now at point B, what are the coordinates of point C?

Jul 12, 2017

The coordinates of point $C$ are $\left(20 , 12\right)$

#### 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

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}4 \\ 6\end{matrix}\right) = \left(\begin{matrix}- 4 \\ - 6\end{matrix}\right)$

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

$\vec{C B} = \frac{1}{2} \vec{C A '}$

$\left(\begin{matrix}8 - x \\ 3 - y\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}- 4 - x \\ - 6 - y\end{matrix}\right)$

So,

$8 - x = \frac{1}{2} \left(- 4 - x\right)$

$16 - 2 x = - 4 - x$

$x = 20$

and

$3 - y = \frac{1}{2} \left(- 6 - y\right)$

$6 - 2 y = - 6 - y$

$y = 12$

Therefore,

point $C = \left(20 , 12\right)$