Question

Points A and B are at `(5 ,9 )` and `(8 ,6 )`, respectively. Point A is rotated counterclockwise about the origin by `pi` and dilated about point C by a factor of `4`. If point A is now at point B, what are the coordinates of point C?

Answer 1

The point `C=(-28/3, -14)`

Explanation:

The matrix of a rotation counterclockwise by `pi` about the origin is

`((-1,0),(0,-1))`

Therefore, the transformation of point `A` is

`A'=((-1,0),(0,-1))((5),(9))=((-5),(-9))`

Let point `C` be `(x,y)`, then

`vec(CB)=4vec(CA')`

`((8-x),(6-y))=4((-5-x),(-9-y))`

So,

`8-x=4(-5-x)`

`8-x=-20-4x`

`3x=-28`

`x=-28/3`

and

`6-y=4(-9-y)`

`6-y=-36-4y`

`3y=-42`

`y=-14`

Therefore,

The point `C=(-28/3,-14)`