Geometry · StudentHub Lesson

The Pythagorean Theorem

Use the Pythagorean theorem to find missing side lengths in right triangles.

20 minIntermediate

What you will learn

State the Pythagorean theorem
Identify the hypotenuse in a right triangle
Calculate a missing side using the theorem
Apply the theorem to real-world distance problems
Recognize when a triangle is a right triangle

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Math Antics - The Pythagorean Theorem · mathantics

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Topic notes

Main Idea

The Pythagorean theorem relates the three sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.

Key Concepts

Only works for right triangles (one 90° angle)
The hypotenuse is the longest side, opposite the right angle
Can be used to find any missing side if the other two are known

Definitions

Hypotenuse: the side opposite the right angle, the longest side
Legs: the two shorter sides forming the right angle
Theorem: a mathematical statement proven true

Formulas

a² + b² = c², where c is the hypotenuse
c = √(a² + b²)
a = √(c² - b²)

Examples

Legs 3 and 4: c = √(9+16) = √25 = 5
Hypotenuse 13, one leg 5: other leg = √(169-25) = √144 = 12

Common Mistakes

Mixing up which side is the hypotenuse
Forgetting to take the square root at the end
Using the theorem on non-right triangles

Key concepts

Right triangle sidesHypotenuse identificationa²+b²=c² formulaSolving for missing sidesReal-world distance applications

Important terms

Hypotenuse: The longest side of a right triangle, opposite the right angle.
Leg: One of the two shorter sides of a right triangle.
Right angle: A 90-degree angle.

Worked examples

Problem

Find the hypotenuse of a right triangle with legs 6 and 8

1. Apply a²+b²=c²
2. 36+64=100
3. c=√100

Answer: c = 10

Problem

A right triangle has hypotenuse 10 and one leg 6. Find the other leg.

1. c² - a² = b²
2. 100-36=64
3. b=√64

Answer: b = 8

Quick revision

Only applies to right triangles
c is always the hypotenuse
a²+b²=c²
Take the square root to finish
Used for distance calculations
Legs are the two shorter sides

Check your understanding

Question 1 · Multiple choice

In a right triangle, which side is the hypotenuse?

Question 2 · Multiple choice

Legs of a right triangle are 5 and 12. What is the hypotenuse?

Question 3 · Short answer

Find the missing leg if the hypotenuse is 17 and one leg is 8.

Question 4 · Short answer

Find the hypotenuse for legs 9 and 12.

Question 5 · True or false

The Pythagorean theorem can be applied to any triangle.

Done with The Pythagorean Theorem?

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