Question

Points A and B are at `(6 ,7 )` and `(7 ,4 )`, respectively. Point A is rotated counterclockwise about the origin by `pi/2` and dilated about point C by a factor of `2`. If point A is now at point B, what are the coordinates of point C?

Answer 1

The coordinates of point `C=(-21,8)`

Explanation:

Point `A=((6),(7))` and point `B=((7),(4))`

The rotation of point `A` counterclockwise tranforms the point `A` into

`A'=((-7),(6))`

Let point `C=((x),(y))`

The dilatation is

`vec(CB)=2vec(CA')`

`((7),(4))-((x),(y))=2*((-7),(6))-((x),(y)))`

Therefore,

`7-x=2(-7-x)`

`7-x=-14-2x`

`x=-14-7=-21`

`4-y=2(6-y)`

`4-y=12-2y`

`y=12-4=8`

The point `C=(-21,8)`