The Pythagorean Theorem is an important theorem, as it comes up often in high school math. It makes the question of how to find the hypotenuse of a right triangle is easy to answer.
Plus, we can use it to find the base of a special right triangle and apply the converse of the Pythagorean Theorem. Let’s review this basic but foundational concept in trigonometry and how it relates to finding the hypotenuse of a right triangle.
Defining the Hypotenuse
First, let's demystify this term. "Hypotenuse" is simply a term that means "the longest side of a right triangle." The hypotenuse is the opposite side of the right angle in the triangle. It’s also the longest side of the triangle.
If a problem asks you to calculate the length of hypotenuse c in a triangle with side a, side b, and hypotenuse c, then you are working with a right-angled triangle.
How the Pythagorean Theorem Applies
The Pythagorean Theorem is a well-known theorem developed by a Greek mathematician named Pythagoras around 500 BC. It related the length of the hypotenuse of a right triangle to the lengths of the legs (the legs of the triangle are the two shorter sides). The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the legs is always equal to the square of the length of the hypotenuse.
The Pythagorean Theorem states that:
Because of the Pythagorean Theorem, it is easy to find the hypotenuse of a right triangle if we are given the sides of a right triangle. In fact, if we know the lengths for any two sides (leg A, leg B, and hypotenuse C) we can easily figure out the missing side by applying the formula.
This can help you find any missing side. But for now, let's see an example where we know the length of the sides and want to find the hypotenuse:
Here we know the length of the side (a = 3 and b = 4). If we substitute it into the formula, we get:
And by taking the square root on both sides, we figure out that the hypotenuse c = 5!
Summary for Finding the Hypotenuse
The Pythagorean Theorem helps us calculate the hypotenuse of a right triangle if we know the sides of the triangle.
There are more advanced trigonometric functions that allow us to calculate the third side of a triangle, even non-right triangles, given a particular degree angle and side length.
But either way, practice applying the Pythagorean Theorem until you feel confident with right triangles. Then, you can learn how to find the third side of any triangle.