Bell Ringer
30-60-90 Triangles
• A Right Triangle with angle measures of 30, 60, and 90 are called 30-60-90 triangles.
Example 1 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.
Take the square root of each side. b = 3
In the diagram, PQR is a 30° –60° –90° triangle with PQ = 2 and PR = 1. Find the value of b.
Example 2 Find Hypotenuse Length
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.
In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse.
ANSWER The length of the hypotenuse is 24.
Example 3 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.
ANSWER The length of the longer leg is 5 . 3
30° –60° –90° Triangle Theoremlonger leg = shorter leg · 3
Substitute.= 5 · 3
SOLUTION
The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times .3
Now You Try Find Lengths in a Triangle
ANSWER 14
Find the value of x. Write your answer in radical form.
1.
2.
3.
ANSWER 3 3
ANSWER 10 3
Example 4 Find Shorter Leg Length
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.
2.9 ≈ x Use a calculator.
ANSWER The length of the shorter leg is about 2.9.
Substitute.5 = x · 3
30° –60° –90° Triangle Theoremlonger leg = shorter leg · 3
SOLUTION
The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times 3.
= x53
Divide each side by .3
Example 5 Find Leg Lengths
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.
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.
Shorter leg
hypotenuse = 2 · shorter leg
Longer leg
longer leg = shorter leg · 3
8 = 2 · x y = 4 · 3
4 = x y = 4 3
Example 5 Find Leg Lengths
ANSWER The length of the shorter leg is 4.The length of the longer leg is 4 .3
Now You Find Leg Lengths
ANSWER 3.5
Find the value of each variable. Round your answer to the nearest tenth.
4.
5.
ANSWER x = 21; y = 21 ≈ 36.43
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