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.

Related Topics

Slope of a LineQuadratic EquationsThe Unit Circle

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.