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.

**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.

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: 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.

**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 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.

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.