# 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.

# 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.