The Pythagorean Theorem

a2 + b2 = c2

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The Pythagorean Theorem is one of the most important ideas in geometry because it connects the three sides of any right triangle. If a triangle has one 90 degree angle, the lengths of its two legs and its hypotenuse follow a simple equation. This theorem is used in construction, navigation, physics, and coordinate geometry. It gives a reliable way to find missing distances when right angles are involved.

In a right triangle, the side opposite the 90 degree angle is called the hypotenuse, and it is always the longest side. If the legs have lengths $a$ and $b$ , and the hypotenuse has length $c$ , then $a^2 + b^2 = c^2$ . You can use this equation to solve for any missing side as long as you know the other two. The theorem also helps check whether a triangle is a right triangle by testing whether its side lengths satisfy the equation.

Key Facts

For any right triangle, $a^2 + b^2 = c^2$
c is the hypotenuse, the side opposite the 90 degree angle
To find the hypotenuse: $c = \sqrt{a^2 + b^2}$
To find a leg: $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$
A 3, 4, 5 triangle is a common Pythagorean triple because $3^2 + 4^2 = 5^2$
In the coordinate plane, distance between points is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Vocabulary

Right triangle: A triangle that has one angle measuring exactly 90 degrees.
Leg: One of the two sides that form the 90 degree angle in a right triangle.
Hypotenuse: The side opposite the 90 degree angle, and the longest side of a right triangle.
Pythagorean triple: A set of three whole numbers that satisfy $a^2 + b^2 = c^2$ .
Distance formula: A formula based on the Pythagorean Theorem that finds the distance between two points on a coordinate plane.

Common Mistakes to Avoid

Using the theorem on a triangle that is not a right triangle, which is wrong because $a^2 + b^2 = c^2$ only works when one angle is 90 degrees.
Calling the wrong side the hypotenuse, which is wrong because the hypotenuse must be opposite the right angle and must be the longest side.
Adding side lengths before squaring, which is wrong because the theorem uses $a^2 + b^2$ , not $(a + b)^2$ .
Forgetting the square root when solving for a side, which is wrong because after finding $c^2$ or $a^2$ you must take the positive square root to get the actual side length.

Practice Questions

1 A right triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.
2 The hypotenuse of a right triangle is 13 m and one leg is 5 m. Find the length of the other leg.
3 A triangle has side lengths 7, 24, and 25. Explain whether it is a right triangle and justify your answer using the Pythagorean Theorem.

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