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Pythagorean Theorem

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Pythagorean Theorem

Study Guide

Key Definition

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$): $a^2 + b^2 = c^2$.

Important Notes

The Pythagorean Theorem only applies to right triangles.
The hypotenuse is always the longest side.
You can use this theorem to find the length of any side when the lengths of the other two sides are known.
The theorem is fundamental in Euclidean geometry.
It helps in calculating distances and in construction.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$a^2$ means a squared

$\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

The Pythagorean Theorem works because it is derived from the properties of right triangles and the concept of area. The theorem can be visualized by comparing the areas of squares constructed on each side of the triangle.

Remember

Always check to make sure the triangle is a right triangle before applying the Pythagorean Theorem.

Quick Reference

Pythagorean Theorem:$a^2 + b^2 = c^2$

Understanding Pythagorean Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: $a^2 + b^2 = c^2$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $a = 3$ and $b = 4$, what is $c$ in a right triangle?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You want to find the shortest path across a park, shaped like a right triangle, with sides $6$ and $8$ meters. A simple right-triangle diagram is shown.

Think About It

Thinking Exercise

Beginner

Think About This

A ladder leaning against a wall forms a right triangle. If the ladder is $10$ meters long and reaches $8$ meters high, find how far the base of the ladder is from the wall. A simple right-triangle diagram is shown.

Challenge Quiz

Single Choice Quiz

Advanced

For a right triangle with $a = 5$ and $c = 13$, find $b$.

Recap

Watch & Learn

Review key concepts and takeaways