Question

Points A and B are at `(9 ,9 )` and `(7 ,8 )`, respectively. Point A is rotated counterclockwise about the origin by `pi` 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

Point `C(-25, -26)`

Explanation:

From `A(9, 9)`
point A will be at `A'(x_a', y_a')=(-9, -9)` after rotation of `pi` whether by counterclockwise or clockwise.

The formula for dilation with center of dilation `C(x_c, y_c)` by a factor `k` is
`x_a''=k(x_a'-x_c)+x_c` and `y_a''=k(y_a'-y_c)+y_c`

where `(x_a'', y_a'')` is the coordinates of the final position of `A`

Given that `B` is at `(7, 8)`, therefore `x_a''=7` and `y_a''=8`

The coordinates of `C(x_c, y_c)` can now be computed with `k=2`
`x_a''=k(x_a'-x_c)+x_c`
`7=2(-9-x_c)+x_c`
`7=-18-2*x_c+x_c`
`7=-18-x_c`
`x_c=-25`

And
`y_a''=k(y_a'-y_c)+y_c`
`8=2(-9-y_c)+y_c`
`8=-18-2y_c+y_c`
`8=-18-y_c`
`y_c=-26`

`C(x_c, y_c)=(-25, -26)`

God bless....I hope the explanation is useful.