A hypotenuse is the longest side of a right triangle. It is the side directly opposite from the right angle, and students first begin learning this term in geometry during middle school years. You can find the length if given either the other two sides of the triangle, or an angle measure and a side length.

## Pythagorean Theorem

In a right triangle, the two sides that create the 90-degree angle are called legs, and the long side that connects them is called the hypotenuse. You can find the length of the hypotenuse from two legs or a leg and an angle measure. The Pythagorean Theorem is a formula used to find the length of any of the sides of a right triangle when given two sides. The formula is usually expressed as **a^2 + b^2 = c^2**, where a and b are the legs, and c is the hypotenuse. If you are given a and b, you can use them and some algebra to find the length of the hypotenuse. Whatever variable labels the hypotenuse, that side will be c in the Pythagorean Theorem formula.

## Plug It In

To solve a right triangle problem, you will always have to find the missing side of a triangle using the other two sides. To find the hypotenuse, plug in the values for a and b. For example, look at at a triangle with side lengths of of 3 and 4. If you plug them into the formula, 3^2 + 4^2 = c^2, and simplify, you get 9 + 16 = c^2. Adding 9+16 gives you 25 = c^2.

## Solve the Equation

Once you have squared the legs and added them together, you still must get c by itself. To get a variable by itself in an equation, apply the cardinal rule of algebra: whatever you do on one side of the equation, you also do on the other. In this case, you need "c" all by itself, as this is the length of the hypoteneuse. Taking the square root of 25 gives you the square root of c^2: c = 5.

## Triple Triangles

Pythagorean Triples are right triangles that have whole number values for each side and can be used to find the hypotenuse of some triangles without doing any calculations. There are many different triples, but the most common are 3-4-5 and 5-12-13 triangles. These side lengths may be factors in larger triangles, but they will always reduce to an even triple. For example, if you have leg lengths of 10 and 24, you could plug them into the equation and take the square root of 10^2 + 24^2. However, if you know your triples, you would note that 10 and 24 are twice 5 and 12, so the hypotenuse must be twice 13, or 26.

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