Special Right Triangles

In this lesson, we will explore special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles have unique properties that allow us to determine the lengths of their sides easily.

45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle where the angles are 45 degrees, 45 degrees, and 90 degrees. The sides opposite the 45-degree angles are equal, and the relationship between the legs and the hypotenuse is given by:

• Legs: $x$
• Hypotenuse: $x\sqrt{2}$

30-60-90 Triangle

A 30-60-90 triangle has angles of 30 degrees, 60 degrees, and 90 degrees. The sides are in the ratio:

• Short leg (opposite the 30-degree angle): $x$
• Long leg (opposite the 60-degree angle): $x\sqrt{3}$
• Hypotenuse: $2x$

Worked Example

Let's find the lengths of the sides of a 30-60-90 triangle where the short leg is 4.

1. Identify the short leg: $x = 4$
2. Calculate the long leg: $x\sqrt{3} = 4\sqrt{3}$
3. Calculate the hypotenuse: $2x = 2(4) = 8$

Thus, the side lengths of the triangle are 4, $4\sqrt{3}$, and 8.

Key Questions

• What is the ratio of the sides in a 45-45-90 triangle?
• If the hypotenuse of a 30-60-90 triangle is 10, what is the length of the short leg?
• How do you find the length of the hypotenuse in a 45-45-90 triangle?
• What is the long leg in a 30-60-90 triangle if the short leg is 5?
• What is the area of a 45-45-90 triangle with legs of length 6?

Keywords

• Special Right Triangles
• 45-45-90 Triangle
• 30-60-90 Triangle
• Ratios
• Trigonometry