- #1

MathDude

- 8

- 0

## Homework Statement

For (x, y) and U, v) in R

^{2}, define (x,y)~(u,v) if x

^{2}+y

^{2}= u

^{2}+v

^{2}. Prove that ~ defines an equivalence relation on R

^{2}and interpret the equivalence classes geometrically.

## Homework Equations

(none)

## The Attempt at a Solution

The first part is easy. I proved transitivity, reflexivity, and symmetry as per the definition of an equivalence. I'm a little confused how to do so geometrically. Since x

^{2}+ y

^{2}can represent the Pythagorean identity, I'm assuming the proof involves triangles. I understand that the properties of an equivalency (trans, reflect, and sym) can resemble triangle congruence and similarity, but I'm not quite so sure how to apply it. Any hints would be helpful.