30-60-90 Right Triangles

Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

In this lesson, we will learn

• the special right triangle called the 30°-60°-90° triangle.
• how to solve problems involving the 30°-60°-90° right triangle

30º-60º-90º Triangles

Another type of special right triangles is the 30°- 60°- 90° triangle. This is right triangle whose angles are 30°, 60°and 90°. The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of .

            

The hypotenuse is always twice the length of the shorter leg (the side facing the 30° angle). The longer leg (the side facing the 60° angle) is times of the shorter leg.

 

Solve problems involving 30º-60º-90º right triangles

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and  inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the  ratio.             

Step 2:  Yes, it is a 30°- 60°- 90° triangle for n = 4

Step 3:  Calculate the third side.

2n = 2×4 = 8

Answer: The length of the hypotenuse is 8 inches.

You can also recognize a 30°- 60°- 90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°- 60°- 90° special right triangle.

A right triangle with a 30° angle or 60° angle must be a 30°- 60°- 90° special right triangle.

Example 2:

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 inches and one of the angles is 30°.

Solution:

Step 1: This is a right triangle with a 30° angle so it must be a 30°- 60°- 90° triangle.

You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio , we get that 2n = 8 Þ n = 4.

Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4 and .

Answer: The lengths of the two sides are 4 inches and inches.

The following videos give more examples of 30-60-90 triangles.

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