Question

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

Answer 1

I get `4 sqrt{2}` for the first endpoint and `0` for the second.

Explanation:

Yikes, that's a lot of transformations. We're just interested in the image of each endpoint.

To do the dilation we start by getting a direction vector from the dilation point to each endpoint, essentially translating the dilation point to the origin.

`(4,1) - (2,2) = (2,-1) quad quad quad quad (2,3)-(2,2)=(0,1)`

We dilate each direction vector by a factor of two and translate back:

`(2,2)+2(2,-1)=(6,0) quad quad quad quad (2,2)+2(0,1)=(2,4)`

Reflecting through `x=-2` leaves the `y` coordinate alone:

`(6,0) to (2-6,0)=(-4,0) quad quad quad quad (2,4) to (2-2,4)=(0,4)`

Reflecting through `y=4` leaves the `x` coordinate alone:

`(-4,0) to (-4,4-0)=(-4,4) quad quad quad quad (0,4) to(0,0)`

If I did that right the first endpoint is `\sqrt{4^2+4^2}=4\sqrt{2}` from the origin and the second endpoint is the origin, so a distance of zero.