# Points A and B are at (3 ,8 ) and (7 ,3 ), 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?

Apr 18, 2018

C=$\left(- \frac{11}{2} , \frac{43}{- 4}\right)$

#### Explanation:

Here Point A=(3,8).
Rotating 'bout the origin by $\pi$ gives $A ' \left(- 3 , - 8\right)$

Again if the point $A '$ is dilated through the center C with scale factor 5 yield it's next point a point $B \left(7 , 3\right)$.

We know dilation of the coordinates are,Let k be the scale factor and (a,b) be the point C.
$x ' = k \left(x - a\right) + a$
$y ' = k \left(y - b\right) + b$

For a,
$x ' = k \left(x - a\right) + a$
$\mathmr{and} , 7 = 5 \left(- 3 - a\right) + a$
$\therefore a = - \frac{11}{2}$

For b,
$y ' = k \left(y - b\right) + b$
$3 = 5 \left(- 8 - b\right) + b$
$\therefore b = - \frac{43}{4}$

Hence $C \left(- \frac{11}{2} , - \frac{43}{4}\right)$ is the required point of dilation.