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# How to Find the Hypotenuse of a Triangle

## Understanding the Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics that describes the relationship between the sides of a right triangle. It states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

The formula for the Pythagorean Theorem is:

cÂ² = aÂ² + bÂ²

Where c is the length of the hypotenuse, and a and b are the lengths of the two legs.

This theorem can be used to find the length of any side of a right triangle if the other two sides are known. In this case, if we want to find the length of the hypotenuse, we can rearrange the formula to solve for c:

c = âˆš(aÂ² + bÂ²)

Where the symbol âˆš represents the square root.

By understanding and applying the Pythagorean Theorem, we can easily find the length of the hypotenuse of any right triangle.

## Identifying the Lengths of the Legs

To find the hypotenuse of a triangle, we first need to identify the lengths of the legs. The legs are the two sides of the triangle that form the right angle.

If the lengths of the legs are not given, we can use the information provided to identify them. For example, if we are given the length of the hypotenuse and one of the angles, we can use trigonometric functions to find the length of the legs.

Alternatively, we can use the Pythagorean Theorem to find the lengths of the legs if the length of the hypotenuse and one of the leg lengths are given. We can rearrange the formula to solve for a or b, depending on which leg length is given:

a = âˆš(cÂ² – bÂ²)

b = âˆš(cÂ² – aÂ²)

Where a and b are the lengths of the legs, and c is the length of the hypotenuse.

By identifying the lengths of the legs, we can apply the Pythagorean Theorem to find the length of the hypotenuse.

## Squaring and Adding the Leg Lengths

Once we have identified the lengths of the legs of a right triangle, we can use the Pythagorean Theorem to find the length of the hypotenuse.

To do this, we need to square the lengths of the legs, add them together, and then take the square root of the sum. This is because the Pythagorean Theorem states that the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

For example, if the lengths of the legs are a = 3 and b = 4, we can use the Pythagorean Theorem to find the length of the hypotenuse as follows:

cÂ² = aÂ² + bÂ²

cÂ² = 3Â² + 4Â²

cÂ² = 9 + 16

cÂ² = 25

c = âˆš25

c = 5

Therefore, the length of the hypotenuse is 5.

By squaring and adding the leg lengths, we can easily find the length of the hypotenuse of any right triangle.

## Taking the Square Root of the Sum

After we have squared and added the lengths of the legs of a right triangle, we need to take the square root of the sum to find the length of the hypotenuse.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 x 5 = 25.

In the context of the Pythagorean Theorem, we take the square root of the sum of the squares of the legs to find the length of the hypotenuse. This is because the Pythagorean Theorem states that the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

For example, if we have squared and added the lengths of the legs of a right triangle and obtained the sum of 25, we can take the square root of 25 to find the length of the hypotenuse:

c = âˆš25

c = 5

Therefore, the length of the hypotenuse is 5.

By taking the square root of the sum of the squares of the legs, we can easily find the length of the hypotenuse of any right triangle.

## Solving Examples with Different Triangles

To find the hypotenuse of a right triangle, we can follow the steps of the Pythagorean Theorem: identify the lengths of the legs, square and add them, and then take the square root of the sum. Let’s apply these steps to solve some examples with different triangles.

Example 1: Find the length of the hypotenuse of a right triangle with legs of length 6 and 8.

cÂ² = aÂ² + bÂ²

cÂ² = 6Â² + 8Â²

cÂ² = 36 + 64

cÂ² = 100

c = âˆš100

c = 10

Therefore, the length of the hypotenuse is 10.

Example 2: Find the length of the hypotenuse of a right triangle with legs of length 5 and 12.

cÂ² = aÂ² + bÂ²

cÂ² = 5Â² + 12Â²

cÂ² = 25 + 144

cÂ² = 169

c = âˆš169

c = 13

Therefore, the length of the hypotenuse is 13.

By applying the Pythagorean Theorem, we can find the length of the hypotenuse of any right triangle, regardless of the lengths of the legs.

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