# Converse of The Pythagorean Theorem

## What is the converse of the Pythagorean theorem?

The Pythagorean theorem allows you to find the length of any side of a right triangle if you know the lengths of the other two sides. It can be viewed in another way, as the **Converse Of The Pythagorean theorem**, to determine if a given triangle is a right triangle just by knowing the lengths of its three sides.

## Pythagorean theorem

To understand the converse of the Pythagorean Theorem, you need to know and recall the Pythagorean Theorem itself:

This formula works for any right triangle **ABC** where * a* and

*are legs and*

**b***is the hypotenuse. The theorem works for all right triangles, so if you know any two lengths (say,*

**c***and*

**a***), you can find the unknown length (in our example,*

**c***). That is a useful application of the Pythagorean theorem.*

**b**## The law of cosines

Before we leap ahead, let's make sure we see the special application of the Law of Cosines in the Pythagorean Theorem.

First, here is the **Law of Cosines** for **△ABC** where aa and bb are legs and cc is the hypotenuse, with **∠C** the right angle opposite the hypotenuse:

The cosine of **90°** is **0**, which leaves you with** 2ab × 0**, so the entire expression, **2ab cos(∠C) = 0** and **can be removed**, leaving just the Pythagorean Theorem.

## Converse of the Pythagorean theorem

Suppose, though, we start at the "other end." We have three sides * a*,

*, and*

**b***, but are not certain*

**c****△ABC**is a right triangle. In that case, we can apply the converse of the Pythagorean theorem, which states:

## Applying the converse of the Pythagorean theorem

Suppose you are given the lengths of three sides of a triangle and asked to determine if it is a right triangle. A common reason for this might be in architecture or in engineering, like getting the correct length of guy-wires bracing an important bridge. Right angles in engineering are very strong.

You know the three sides (legs * a* and

*, hypotenuse*

**b***) are as follows:*

**c**If ${a}^{2}+{b}^{2}={c}^{2}$, then the triangle has to be a right triangle and the guy-wires have to be the perfect and safe length to hold up the bridge.

So you put the three lengths into the Pythagorean Theorem formula:

What a relief! The guy-wires are all the correct length to keep the bridge at a safe, sturdy right angle.

## Lesson summary

Today you reviewed what the Pythagorean theorem is and why it is useful, how to write and use the formula for the Pythagorean theorem (${a}^{2}+{b}^{2}={c}^{2}$), and learned how the Pythagorean theorem is one application of the law of cosines.

You also learned what the converse of the Pythagorean theorem is; namely, any triangle in which the square of the longest side of a triangle is equal to the sum of the squares of the other two sides must be a right triangle.