Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10: 9781285195698
ISBN 13: 978-1-28519-569-8

Chapter 5 - Section 5.4 - The Pythagorean Theorem - Exercises - Page 241: 34


Using proof by contradiction Suppose $ a^{2}+b^{2}= c^{2} $ Then by The pythagorean theorem the triangle is a right triangle this contradicts the given that triangle ABC is not a right triangle. Therefore $a^{2}+ b^{2} \ne c^{2} $.

Work Step by Step

We know from given theorems that if $a^2 + b^2 < c^2$, then the triangle is obtuse and that if $a^2 + b^2 > c^2$, then the triangle is acute. The only other triangle is a right triangle, and the only other formula is $a^2 + b^2 = c^2$. Thus, a triangle is right if and only if $a^2 + b^2 = c^2$. Thus, if $a^2+b^2 \ne c^2$, then the triangle is not a right triangle.
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