Geometry

Pythagorean Theorem

Big Picture

Why don’t we see mathematicians at the beach very often? They don’t need the sun because they have sine and cosine to give them a tan! If you don’t get this joke, you will after reading this study guide. Right triangles play a large role in trigonometry. The Pythagorean theorem and many trigonometric ratios depend on the properties of right triangles.

Key Terms

Triangle: A closed figure made by three line segments intersecting at their endpoints.

Vertex (plural, vertices): The point where the segments intersect.

Right Triangle: A triangle that contains one right angle.

Acute Triangle: A triangle with an acute angle.

Obtuse Triangle: A triangle with an obtuse angle.

Hypotenuse: Longest leg in a right triangle, opposite of the 90° angle.

Pythagorean Triple: Three integers that satisfy the Pythagorean Theorem.

Naming Sides and Angles

The sides and angles of triangles are generally named as shown in the picture below.

A, B, and C are vertices.
a, b, and c are sides. a is opposite ∠A, b is opposite ∠B, and c is opposite ∠C.
It is very important to always name triangles using capital letters for the vertices and corresponding lowercase letters for the opposite sides - that’s how they’re named in trigonometric formulas, theorems, and problems.

Pythagorean Theorem

Pythagorean Theorem: For a right triangle with legs of lengths a and b and hypotenuse of length c, then:
\(\color{blue}{a}\color{black}{^2 + }\color{blue}{b}\color{black}{^2 = }\color{red}{c}\color{black}{^2}\)

The largest length will ALWAYS be the hypotenuse.

Given the lengths of two sides of a right triangle, the Pythagorean Theorem can be used to find the third side length of the triangle.

Geometry

Pythagorean theorem Cont.

Pythagorean Theorem (cont.)

Pythagorean Triples

A Pythagorean triple are three positive integers that make the Pythagorean Theorem true.

One example of a Pythagorean triple is the lengths 3, 4,and 5. Other Pythagorean triples can result by multiplying each integer in the triple by the same factor. For example, 3×2, 4×2, 5×2 = 6, 8, 10, which is another Pythagorean triple. This gives us an infinite number of Pythagorean triples. Some of the frequently used Pythagorean triples are:

3, 4, 5
5, 12, 13
7, 24, 25
8, 15, 17

The largest length must be the hypotenuse - if a triangle has two legs of lengths 3 and 5, the hypotenuse is not 4!

Tip: It is helpful to memorize these frequently used triples.

Converse of Pythagorean Theorem

Converse of the Pythagorean Theorem: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If \(c^2 = a^2 + b^2\) then the triangle is a right triangle

The theorem can be extended to determine if a triangle is obtuse or acute.

Theorem: If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the triangle is acute.

Theorem: If the sum of the squares of the two shorter sides is less than the square of the longest side, then the tri-angle is obtuse.

Using Side Lengths to Classify Triangles by Angles

If c² = a² + b², then m ∠C = 90º
and △ABC is a right triangle.

If c² < a² + b², then m ∠C < 90º
and △ABC is an acute triangle.

If c² > a² + b², then m ∠C > 90º
and △ABC is an obtuse triangle.

Proving the Distance Formula

The Pythagorean Theorem can also be used to prove the distance formula, which is:
\(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\)

1. Start with a line segment of unknown length (d) whose endpoints are
at \((x_1, y_1)\) and \((x_2, y_2)\)

2. Draw the horizontal and vertical lengths to create a right triangle using the line segment as the hypotenuse.