Question

A line segment goes from `(3 ,2 )` to `(1 ,3 )`. The line segment is dilated about `(1 ,1 )` by a factor of `2`. Then the line segment is reflected across the lines `x=4` and `y=-3`, in that order. How far are the new endpoints from the origin?

Answer 1

`color(blue)("Distances of A & B from origin after dilation and reflection " 9.49, 13.04 " respectively"`

Explanation:

`A (3,2), B (1,3), " dilated by 2 about C(1,1)"`

`A(x,y) -> A'(x,y) = 2 * A(x,y) - C(x,y) = (2*(3,2)-(1,1)) = (5, 3)`

`B(x,y) -> B'(x,y) = 2 * B(x,y) - C(x,y) = (2*(1,3)-(1,1)) = (1, 5)`

Line segment A'B' reflected across x = 4, y = -3, in that order.

`color(crimson)("reflect thru x = 4, y = -3 ", h=4, k= -3. (2h-x, 2k-y)"`

`A''(x, y) = (2h - x, 2k-y) = (2*4 - 5, 2*-3 - 3) = (3, -9)`

`B''(x, y) = (2h - x, 2k-y) = (2*4 - 1, 2*-3 - 5) = (7, -11)`

`OA'' = sqrt(3^2 + 9^2) = 9.49`

`OB'' = sqrt(7^2 + 11^2) = 13.04`