The Converse Of Pythagorean Theorem - Varsity Tutors

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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 triangleRemember 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

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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.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

AYes, it is a right triangleBNo, it is not a right triangleCCannot determineDNone of the aboveCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

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.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Quick Quiz

Single Choice QuizBeginner

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

AYes, it is a right triangleBNo, it is not a right triangleCIt is an equilateral triangleDNone of the aboveCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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