45-45-90 Triangle Theorem

This page explains the fundamental properties of the 45-45-90 triangle, which is a right triangle with two 45-degree angles.

Highlight: In a 45-45-90 triangle, both legs are equal in length, and the hypotenuse is √2 times the length of a leg.

Example: If a leg is length S, then the other leg is also S, and the hypotenuse is S√2.

30-60-90 Triangle Theorem

This page details the characteristics of the 30-60-90 triangle and its side length relationships.

Definition: A 30-60-90 triangle is a right triangle with angles of 30, 60, and 90 degrees.

Highlight: The hypotenuse is always twice the length of the shorter leg, while the longer leg is √3 times the shorter leg.

Important Note on Applications

This page emphasizes the significance of memorizing these special triangles.

Quote: "These two triangles are very important to memorize. We will use them all semester and they are frequently used in other math classes."

Examples Introduction

This page serves as a transition to practical applications of the theorems.

Vocabulary: Examples demonstrate how to apply theoretical knowledge to solve real mathematical problems.

Example 1: 30-60-90 Triangle

This page demonstrates how to find missing sides in special right triangles when given the hypotenuse of a 30-60-90 triangle.

Example: Given a hypotenuse of 4 units, the short side (opposite to 30°) is 2 units, and the long side (opposite to 60°) is 2√3 units.

Example 2: 45-45-90 Triangle

This page shows how to solve for missing sides in a 45-45-90 triangle when given one leg.

Example: When one leg is 9 units, the other leg is also 9 units, and the hypotenuse is 9√2 units.

Example 3: Part 1

This page begins a multi-step example solving a 30-60-90 triangle with a known long side of 18 units.

Highlight: The solution process begins by using the relationship s√3 = 18 to find the value of s.

Example 3: Part 2

This page continues the solution process, focusing on rationalizing denominators.

Example: The short side $x$ is calculated to be 6√3 units through rationalization of the denominator.

Example 3: Part 3

This page concludes the example by finding the hypotenuse.

Highlight: The hypotenuse $y$ is calculated as 12√3 units by doubling the short side length.

Special Right Triangles Introduction

This page introduces the concept of special right triangles, which are essential geometric figures with unique properties that make calculating side lengths more efficient.

Definition: Special right triangles are right triangles with specific angle measurements that create consistent relationships between their sides.