Master

The Converse of Pythagorean Theorem

Master the converse of pythagorean theorem with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

The Converse of Pythagorean Theorem

Study Guide

Key Definition

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle: $c^2 = a^2 + b^2$

Important Notes

The converse of the Pythagorean theorem helps identify right triangles.
If $c^2 = a^2 + b^2$, then the triangle is right-angled.
Use the longest side as $c$ when applying the converse.
The converse is a specific case of the Pythagorean theorem.
Triangles that satisfy $c^2 = a^2 + b^2$ are right triangles.

Mathematical Notation

$c^2$ represents the square of the length of the longest side

$a^2 + b^2$ represents the sum of the squares of the other two sides

$\angle$ represents an angle

$\triangle$ represents a triangle

Remember to use proper notation when solving problems

Why It Works

Proof Sketch: Let triangle ABC have sides of lengths a, b, and c (with c the longest side) and suppose $c^2 = a^2 + b^2$. Construct a right triangle DEF with legs $DE = a$ and $DF = b$; by the Pythagorean theorem, its hypotenuse EF satisfies $EF^2 = DE^2 + DF^2 = a^2 + b^2 = c^2$, so EF = c. By the Side-Side-Side postulate, ∆ABC ≅ ∆DEF, implying ∠C = ∠D = 90°. Therefore, triangle ABC is right-angled at C.

Remember

Always check if $c^2 = a^2 + b^2$ to determine if a triangle is right-angled.

Quick Reference

Converse of Pythagorean Theorem:$c^2 = a^2 + b^2 \Rightarrow \triangle \text{is right-angled}$

Understanding The Converse of Pythagorean Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The converse of the Pythagorean theorem states that if in a triangle the square of the longest side equals the sum of the squares of the other two sides ($c^2 = a^2 + b^2$), then the triangle must have a right angle opposite the longest side. Simply verify this equation to determine if a given triangle is right-angled.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Check if a triangle with sides $6$, $8$, and $10$ is a right triangle.

Real-World Problem

Question Exercise

Intermediate

Converse Application

A triangular park gate has side lengths 7 ft, 24 ft, and 25 ft. Use the converse of the Pythagorean theorem to determine whether the gate is right-angled.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a triangle with sides $5$, $12$, and $13$, prove whether it is a right triangle using the converse of the Pythagorean theorem.

Quick Quiz

Single Choice Quiz

Beginner

A triangle has sides $9$, $12$, and $15$. Is it a right triangle?

Recap

Watch & Learn

Review key concepts and takeaways