The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right triangle. One of the most common applications of this theorem is finding the length of the hypotenuse, which is the side opposite the right angle. In this guide, we will walk you through the step-by-step process of finding the hypotenuse using the Pythagorean Theorem.

**Understanding the Pythagorean Theorem**

Before we delve into finding the hypotenuse, let’s first understand the Pythagorean Theorem. In a right triangle, where one angle is 90 degrees, the sum of the squares of the two shorter sides (legs) is equal to the square of the length of the hypotenuse. Mathematically, it can be expressed as:

a² + b² = c²

Where:

– a and b are the lengths of the two legs

– c is the length of the hypotenuse

**Identifying Sides in a Right Triangle**

To find **the hypotenuse**, you need to identify which sides are given and which side you are trying to find. In this case, since we are looking for **the hypotenuse**, it will be represented by ‘c’ in our equation. The other two sides (legs) will be ‘a’ and ‘b’.

**Applying Pythagorean Theorem**

Once you have identified **the sides** in your right triangle, you can now apply **the Pythagorean Theorem**. Simply substitute **the lengths** of **the legs (a and b)** into **the formula**:

a² + b² = c²

After substituting values into **the equation**, solve for ‘c’ by taking **the square root** of both sides:

c = √(a² + b²)

This calculation will give you **the length** of **the hypotenuse**.

**Example Calculation**

Let’s consider an example where one leg has a length of 3 units and another leg has a length of 4 units. To find out how long **the hypotenuse** is, we can plug these values into our formula:

c = √(3² + 4²)

c = √(9 + 16)

c = √25

c = 5

Therefore, in this example, **the length** of **the hypotenuse** would be 5 units.

By following these steps and understanding how to apply **the Pythagorean Theorem**, you can confidently find out how long **the hypotenuse** is in any right triangle scenario.