Question

A line segment goes from `(5 ,2 )` to `(4 ,2 )`. The line segment is reflected across `x=-3`, reflected across `y=-5`, and then dilated about `(2 ,0 )` by a factor of `2`. How far are the new endpoints from the origin?

Answer 1

Distance of new points from origin

`vec(OA”) = 33.94`

`vec(OB”) = 32.56`

Explanation:

Given points `A(5,2) , B (4,2)`

Reflected across `x = -3`, `y = -5`

![Given points `A(5,2) , B (4,2)`

Reflected across `x = -3`, `y = -5`


`A (x,y) -> A’(x, y) = (5,2) -> ((2x’ - 5), (2y’ - 2)) => (-11, -12)`

`A’(x,y) => ((-6 - 5), (-10 - 2)) => (-11, -12)`

`B (x,y) - > B’ (x,y) = (4 , 2) -> ((2x’ - 4), (2y’ - 2))`

`B’(x,y) => ((-6 - 4), (-10 - 2)) => (-10, -12)`

New coordinates after reflection are `A’(-11, -12), B’(-10, -12)`

Now we we have to find A”, B” after rotation about point C (2,0) with a dilation factor of 2.

`A”(x, y) -> 2 * A’(x,y) - C (x,y)`

`A”((x),(y)) = 2 * ((-11), (-12)) - ((2), (0)) = ((-24),(-24))`

Similarly, `B”((x),(y)) = 2 * ((-10), (-12)) - ((2), (0)) = ((-22),(-24))`

New endpoints are `A”(-24, -24), B”(-22, -24)`

Distance of new points from origin

`vec(OA”) = sqrt(-24^2 + -24^2) = 33.94`

`vec(OB”) = sqrt(-22^2 + -24^2) = 32.56`]

`A (x,y) -> A’(x, y) = (5,2) -> ((2x’ - 5), (2y’ - 2) => (-11, -12)`

`A’(x,y) => ((-6 - 5), (-10 - 2) => (-11, -12)`

`B (x,y) - > B’ (x,y) = (4 , 2) -> ((2x’ - 4), (2y’ - 2)`

`B’(x,y) => ((-6 - 4), (-10 - 2) => (-10, -12)`

New coordinates after reflection are `A’(-11, -12), B’(-10, -12)`

Now we we have to find A”, B” after rotation about point C (2,0) with a dilation factor of 2.

`A”(x, y) -> 2 * A’(x,y) - C (x,y)`

`A”((x),(y)) = 2 * ((-11), (-12)) - ((2), (0)) = ((-24),(-24))`

Similarly, `B”((x),(y)) = 2 * ((-10), (-12)) - ((2), (0)) = ((-22),(-24))`

New endpoints are `A”(-24, -24), B”(-22, -24)`

Distance of new points from origin

`vec(OA”) = sqrt(-24^2 + -24^2) = 33.94`

`vec(OB”) = sqrt(-22^2 + -24^2) = 32.56`