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Pythagorean Theorem

Discover the most famous equation in geometry — from right triangles to the distance formula

Over 2,500 years ago, the Greek mathematician Pythagoras discovered something remarkable about right triangles: if you square the two shorter sides and add them together, you always get the square of the longest side. That relationship — a² + b² = c² — is one of the most useful equations in all of mathematics.

The longest side of a right triangle (the one opposite the right angle) is called the hypotenuse. The Pythagorean theorem lets you find any missing side of a right triangle if you know the other two. It also leads to the distance formula, which tells you the distance between any two points on a coordinate plane.

In this lesson, you'll explore a classic 3-4-5 right triangle on the graph, verify the theorem with real numbers, discover other Pythagorean triples, and connect it all to the distance formula — with an AI tutor guiding you step by step.

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FAQ

What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. For example, a triangle with sides 3, 4, and 5 satisfies 3² + 4² = 9 + 16 = 25 = 5².
How do I use the Pythagorean theorem to find a missing side?
If you know two sides of a right triangle, you can find the third. To find the hypotenuse: c = \sqrt{a^2 + b^2}. To find a leg: a = \sqrt{c^2 - b^2}. For example, if the legs are 6 and 8, then c = √(36 + 64) = √100 = 10.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². The most common ones are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a Pythagorean triple is also a triple — for example, (6, 8, 10) is 2 × (3, 4, 5).
What is the distance formula and how does it relate to the Pythagorean theorem?
The distance formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} calculates the distance between two points. It's derived directly from the Pythagorean theorem: the horizontal difference is one leg, the vertical difference is the other leg, and the distance is the hypotenuse.

Related Topics

Slope of a LineQuadratic EquationsThe Unit Circle