# How to Find Hypotenuse: A Comprehensive Guide for Zeromedia

Halo, Zeromedia! If you are struggling with finding the hypotenuse of a right triangle, you have come to the right place. In this article, we will cover everything you need to know about hypotenuse, including how to find it using various methods and formulas.

## Understanding Hypotenuse

Before we dive into how to find the hypotenuse, let’s make sure we understand what it is. The hypotenuse is the longest side of a right triangle and is always opposite the right angle. You can identify the hypotenuse by looking for the side that is opposite to the right angle, which is usually labeled with a “C” on the triangle diagram.

## Using Pythagorean Theorem to Find Hypotenuse

1. Identify the two legs of the right triangle. The legs are the two shorter sides that form the right angle.
2. Write down the Pythagorean Theorem formula: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.
3. Plug in the values of a and b into the formula.
4. Solve for c by taking the square root of both sides of the equation.

For example, let’s say you have a right triangle whose legs are 3 and 4 units long. To find the length of the hypotenuse, you would use the Pythagorean Theorem as follows:32 + 42 = c29 + 16 = c225 = c2c = √25c = 5Therefore, the length of the hypotenuse of the right triangle with legs of 3 and 4 units is 5 units.

## Using Trigonometric Ratios to Find Hypotenuse

Another way to find the hypotenuse of a right triangle is to use trigonometric ratios. Trigonometric ratios are ratios of the sides of a right triangle and can be used to find the length of a missing side. The three trigonometric ratios are sine, cosine, and tangent.

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### Finding Hypotenuse Using Sine

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, to find the length of the hypotenuse, you can use the formula:

sin θ = opposite / hypotenuse

Rearranging the formula, you get:

hypotenuse = opposite / sin θ

For example, let’s say you have a right triangle whose opposite side to the angle θ is 4 units long, and the angle θ is 30 degrees. To find the length of the hypotenuse, you would use the sine formula as follows:

hypotenuse = 4 / sin 30

Using a calculator, you find that sin 30 is 0.5.

Therefore,

hypotenuse = 4 / 0.5

hypotenuse = 8

Therefore, the length of the hypotenuse of the right triangle whose opposite side to the angle θ is 4 units long, and the angle θ is 30 degrees is 8 units.

### Finding Hypotenuse Using Cosine

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. Therefore, to find the length of the hypotenuse, you can use the formula:

cos θ = adjacent / hypotenuse

Rearranging the formula, you get:

hypotenuse = adjacent / cos θ

For example, let’s say you have a right triangle whose adjacent side to the angle θ is 3 units long, and the angle θ is 60 degrees. To find the length of the hypotenuse, you would use the cosine formula as follows:

hypotenuse = 3 / cos 60

Using a calculator, you find that cos 60 is 0.5.

Therefore,

hypotenuse = 3 / 0.5

hypotenuse = 6

Therefore, the length of the hypotenuse of the right triangle whose adjacent side to the angle θ is 3 units long, and the angle θ is 60 degrees is 6 units.

### Finding Hypotenuse Using Tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, to find the length of the hypotenuse, you can use the formula:

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tan θ = opposite / adjacent

Rearranging the formula, you get:

hypotenuse = opposite / tan θ

For example, let’s say you have a right triangle whose opposite side to the angle θ is 5 units long, and the angle θ is 45 degrees. To find the length of the hypotenuse, you would use the tangent formula as follows:

hypotenuse = 5 / tan 45

Using a calculator, you find that tan 45 is 1.

Therefore,

hypotenuse = 5 / 1

hypotenuse = 5

Therefore, the length of the hypotenuse of the right triangle whose opposite side to the angle θ is 5 units long, and the angle θ is 45 degrees is 5 units.

## Using Special Right Triangles to Find Hypotenuse

Special right triangles are right triangles whose side lengths follow specific patterns. There are two types of special right triangles: 45-45-90 triangles and 30-60-90 triangles. By knowing the patterns of these triangles, you can easily find the length of the hypotenuse.

### Finding Hypotenuse of a 45-45-90 Triangle

A 45-45-90 triangle is a right triangle in which the two legs are congruent. The hypotenuse of a 45-45-90 triangle is equal to the length of the legs multiplied by the square root of 2.

For example, let’s say you have a 45-45-90 triangle whose legs are 3 units long. To find the length of the hypotenuse, you would use the following formula:

hypotenuse = leg × √2

Therefore,

hypotenuse = 3 × √2

hypotenuse = 4.24

Therefore, the length of the hypotenuse of the 45-45-90 triangle whose legs are 3 units long is approximately 4.24 units.

### Finding Hypotenuse of a 30-60-90 Triangle

A 30-60-90 triangle is a right triangle in which the angles measure 30, 60, and 90 degrees. The hypotenuse of a 30-60-90 triangle is equal to twice the length of the shorter leg, or the length of the longer leg multiplied by the square root of 3.

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For example, let’s say you have a 30-60-90 triangle whose shorter leg is 2 units long. To find the length of the hypotenuse, you would use the following formula:

hypotenuse = 2 × √3

Therefore,

hypotenuse = 2 × 1.73

hypotenuse = 3.46

Therefore, the length of the hypotenuse of the 30-60-90 triangle whose shorter leg is 2 units long is approximately 3.46 units.