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30-60-90 Triangles

- A Right Triangle with angle measures of 30, 60, and 90 are called 30-60-90 triangles.

In the diagram, PQR is a 30°–60°–90°triangle with PQ = 2 and PR = 1. Find the value of b.

3

b =

Take the square root of each side.

Find Leg Length

SOLUTION

You can use the Pythagorean Theorem to find the value of b.

(leg)2 + (leg)2 = (hypotenuse)2

Write the Pythagorean Theorem.

12 + b2 = 22

Substitute.

1 + b2 = 4

Simplify.

b2 = 3

Subtract 1 from each side.

Find Hypotenuse Length

In the 30°–60°–90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse.

SOLUTION

The hypotenuse of a 30° –60° –90° triangle is twice as long as the shorter leg.

hypotenuse = 2 · shorter leg

30° –60° –90° Triangle Theorem

= 2 · 12

Substitute.

= 24

Simplify.

The length of the hypotenuse is 24.

ANSWER

SOLUTION

The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times .

longer leg = shorter leg ·

30° –60°–90° Triangle Theorem

= 5 ·

Substitute.

3

3

3

3

The length of the longer leg is 5 .

ANSWER

Find Longer Leg Length

In the 30°–60°–90° triangle at the right, the length of the shorter leg is given. Find the length of the longer leg.

3

ANSWER

3

3

10

ANSWER

Find Lengths in a Triangle

Find the value of x. Write your answer in radical form.

1.

14

ANSWER

2.

3.

In the 30°–60°–90° triangle at the right, the length of the longer leg is given. Find the length x of the shorter leg. Round your answer to the nearest tenth.

SOLUTION

The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times

.

longer leg = shorter leg ·

30°–60°–90° Triangle Theorem

5 = x·

Substitute.

3

3

3

3

5

= x

Divide each side by .

3

Find Shorter Leg Length

The length of the shorter leg is about 2.9.

ANSWER

2.9 ≈ x

Use a calculator.

In the 30°–60°–90° triangle at the right, the length of the hypotenuse is given. Find the length x of the shorter leg and the length y of the longer leg.

Shorter leg

Longer leg

3

3

3

longer leg = shorter leg ·

hypotenuse = 2 · shorter leg

y = 4 ·

y = 4

Find Leg Lengths

SOLUTION

Use the 30° –60° –90° Triangle Theorem to find the length of the shorter leg. Then use that value to find the length of the longer leg.

8 = 2 ·x

4 = x

The length of the shorter leg is 4.

ANSWER

The length of the longer leg is 4 .

3

Find Leg Lengths

3

x = 21;y = 21 ≈ 36.4

ANSWER

Find Leg Lengths

Find the value of each variable. Round your answer to the nearest tenth.

4.

3.5

ANSWER

5.

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