Question

Points A and B are at `(9 ,3 )` and `(4 ,8 )`, respectively. Point A is rotated counterclockwise about the origin by `pi` 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?

Answer 1

The coordinates of point `C` are `=(-31/2,-17/2)`

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))((9),(3))=((-9),(-3))`

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

`vec(CB)=3vec(CA')`

`((4-x),(8-y))=3((-9-x),(-3-y))`

So,

`4-x=3(-9-x)`

`4-x=-27-3x`

`2x=-31`

`x=-31/2`

and

`8-y=3(-3-y)`

`8-y=-9-3y`

`2y=-17`

`y=-17/2`

Therefore,

The point `C=(-31/2,-17/2)`