Chapter 11, Problem 23CUR

To determine

To Prove: No triangle has side lengths of $x,y,x+y$ (using indirect proof).

Explanation of Solution

Given:

A triangle.

Let it is true that the triangle has sides of length $x,y,x+y$ .

Considering the two sides of triangle $x,y$ .

The range of length of third side, $z$ of the triangle:

  $|x-y|<z<x+y$

i.e. maximum value that the third side can have is lesser than $x+y$ .

But it has been assumed that third side is equal to $x+y$ .

Therefore, the assumption is wrong.

Hence proved that, no triangle has side lengths of $x,y,x+y$ (using indirect proof).