Question

A line segment has endpoints at (-1, 4) and (4, 1). What reflection will produce an image with endpoints at (-4,1) and (-1,-4)?

Answer 1

`color(blue)("Reflection in " y=-x)`

Explanation:

Let `A=(-1,4)` and `B=(4,1)`

If we look at the mapping we see that:

`A->A'=(-1,4)->(-4,1)`

So:

`(x,y)->(-y,-x)`

This is a reflection in the line `y=-x`.

We can see this if we reflect the axes in this line. The positive x axis becomes the negative y axis and the positive y axis becomes the negative x axis.

We test this:

`A=(-1,4)->(-y,-x)=(-4,-(-1))=(-4,1)`

`B=(4,1)->(-y,-x)=(-1,-4)`

So this is a reflection in the line:

`y=-x`