# Points A and B are at (4 ,6 ) and (7 ,5 ), respectively. Point A is rotated counterclockwise about the origin by pi/2  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?

Jun 8, 2018

color(green)("Coordinates of " C = (5,(3/2))

#### Explanation:

$A \left(4 , 6\right) , B \left(7 , 5\right) , \text{counterclockwise rotation }$pi/2$, \text{dilation factor} \frac{1}{2}$

New coordinates of A after $\frac{3 \pi}{2}$ counterclockwise rotation

$A \left(4 , 6\right) \rightarrow A ' \left(- 6 , 4\right)$

$\vec{B C} = \left(\frac{1}{2}\right) \vec{A ' C}$

$b - c = \left(\frac{1}{2}\right) a ' - \left(\frac{1}{2}\right) c$

$\left(\frac{1}{2}\right) c = - \left(\frac{1}{2}\right) a ' + b$

$\left(\frac{1}{2}\right) C \left(\begin{matrix}x \\ y\end{matrix}\right) = - \left(\frac{1}{2}\right) \left(\begin{matrix}- 6 \\ 4\end{matrix}\right) + \left(\begin{matrix}7 \\ 5\end{matrix}\right) = \left(\begin{matrix}10 \\ 3\end{matrix}\right)$

color(green)("Coordinates of " 2 *C ((10),3) = C(5,(3/2))