Unit 7 · Topic 1

The Pythagorean Theorem

Overview

For a right triangle, a² + b² = c².

Topic 1 of 2~22 min
Next: Volume of Cylinders, Cones & SpheresAll units
Unit overview

The lesson

Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.

The relationship

For any right triangle with legs a and b and hypotenuse c (the side opposite the right angle): a² + b² = c².

Worked example

A right triangle has legs 3 and 4. Find the hypotenuse.

  1. 1a² + b² = c²: 3² + 4² = c².
  2. 29 + 16 = 25, so c² = 25.
  3. 3c = √25 = 5.

Why this matters

For a right triangle, a² + b² = c².

These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.

Common mistakes to avoid

Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.

Re-reading the question too quickly. Underline what you are solving for before you start calculating.

Key ideas from this lesson

  1. The relationship

Video walkthroughs

Math Antics

The Pythagorean Theorem

Use a² + b² = c² to find a missing side.

Watch on YouTube
Khan Academy

Pythagorean Theorem Intro

Another walkthrough of the theorem with examples.

Watch on YouTube
Khan Academy

Area of an Isosceles Triangle

Use the Pythagorean theorem to find height, then area.

Watch on YouTube
Khan Academy

Introduction to Square Roots

Connect square roots to the Pythagorean theorem.

Watch on YouTube

Practice

For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.

Exercise 1

Try it yourself

A right triangle has legs 5 and 12. Find the hypotenuse.

Step-by-step solution

  1. 15² + 12² = c² → 25 + 144 = 169.
  2. 2c² = 169, so c = √169 = 13.
  3. 3Hypotenuse: 13.

Exercise 2

Try it yourself

A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

Step-by-step solution

  1. 113² − 5² = b² → 169 − 25 = 144.
  2. 2b = √144 = 12.
  3. 3Missing leg: 12.

Exercise 3

Try it yourself

Do sides 7, 24, and 25 form a right triangle?

Step-by-step solution

  1. 17² + 24² = 49 + 576 = 625.
  2. 225² = 625.
  3. 3Yes, it is a right triangle.

Exercise 4

Try it yourself

A ladder leans against a wall. The base is 6 ft from the wall and the ladder is 10 ft long. How high does it reach?

Step-by-step solution

  1. 1Legs: 6 (ground) and h (wall); hypotenuse 10.
  2. 26² + h² = 10² → 36 + h² = 100.
  3. 3h² = 64, h = 8 ft.

Exercise 5

Try it yourself

Find the diagonal of a rectangle with length 9 and width 40.

Step-by-step solution

  1. 19² + 40² = 81 + 1600 = 1681.
  2. 2√1681 = 41.
  3. 3Diagonal length: 41.

Re-read the lesson section if you get stuck, then try again without peeking at the solution.

Khan Academy

Pythagorean theorem

Open resource
Math Worksheets 4 Kids

Pythagorean theorem worksheets

Open resource

Quick check

Answer all questions. Retake the quiz until you feel confident before moving on.

The Pythagorean Theorem

Question 1 of 4

Easy

A right triangle has legs 6 and 8. What is the hypotenuse?

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