# Points A and B are at (9 ,4 ) and (1 ,5 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  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?

Aug 12, 2017

The coordinates of $C = \left(\frac{11}{2} , - 16\right)$

#### Explanation:

Point $A = \left(\begin{matrix}9 \\ 4\end{matrix}\right)$ and point $B = \left(\begin{matrix}1 \\ 5\end{matrix}\right)$

The rotation of point $A$ counterclockwise by $\left(\frac{3}{2} \pi\right)$ tranforms the point $A$ into

$A ' = \left(\begin{matrix}4 \\ - 9\end{matrix}\right)$

Let point $C = \left(\begin{matrix}x \\ y\end{matrix}\right)$

The dilatation is

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

((1),(5))-((x),(y))=3*((4),(-9))-((x),(y)))

Therefore,

$1 - x = 3 \left(4 - x\right)$

$1 - x = 12 - 3 x$

$2 x = 12 - 1 = 11$, $\implies$, $x = \frac{11}{2}$

$5 - y = 3 \left(- 9 - y\right)$

$5 - y = - 27 - 3 y$

$2 y = - 27 - 5 = - 32$, $\implies$, $y = - 16$

The point $C = \left(\frac{11}{2} , - 16\right)$