Triangles and the Pythagorean Theorem Study Guide

Introduction to Triangles and the Pythagorean Theorem

Lesson Summary

In this lesson, you will learn the special names for the sides of a right triangle. You will also learn how to use the Pythagorean theorem to find missing parts of a right triangle and to determine whether three segments will make a right triangle.

The right triangle is very important in geometry because it can be used in so many different ways. The Pythagorean theorem is just one of the special relationships that can be used to help solve problems and find missing information. Right triangles can be used to find solutions to problems involving figures that are not even polygons.

Parts of a Right Triangle

In a right triangle, the sides that meet to form the right angle are called the legs. The side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest of the three sides. It is important that you can correctly identify the sides of a right triangle, regardless of what position the triangle is in.

Review of Squares and Square Roots

Before you study the Pythagorean theorem, let's first review squares and square roots. Just like addition and subtraction are inverses, so are squares and square roots. In other words, they "undo" each other. To square a number, you multiply it by itself. For example, 5² means two factors of five, or five times five, which is 25. Written algebraically, it looks like this: 5² = 5 × 5 = 25. A common mistake is to say that you multiply by two, since two is the exponent (the small raised number). But the exponent tells you how many times to use the base (bottom number) as a factor.

Twenty-five is a perfect square. It can be written as the product of two equal factors. It would be helpful for you to learn the first 16 perfect squares. When completed, the following chart will be a useful reference. It is not necessarily important that you memorize the chart, but you need to understand how the numbers are generated. Even the most basic calculators can help you determine squares and square roots of larger numbers.

The Pythagorean Theorem

The Pythagorean theorem is one of the most famous theorems in mathematics. The Greek mathematician Pythagoras (circa 585–500 b.c.) is given credit for originating it. Evidence shows that it was used by the Egyptians and Babylonians for hundreds of years before Pythagoras.

The Pythagorean theorem can be used to solve many real-life problems. Any unknown length can be found if you can make it a part of a right triangle. You need to know only two of the sides of a right triangle to find the third unknown side. A common mistake is always adding the squares of the two known lengths. You add the squares of the legs only when you are looking for the hypotenuse. If you know the hypotenuse and one of the legs, then you subtract the square of the leg from the square of the hypotenuse. Another common mistake is forgetting to take the square root as your final step. You just need to remember that you are solving not for the square of the side, but for the length of the side.