1. Bell Ringer

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

3. You can use the Pythagorean Theorem to find the value of b.
Example 1
In the diagram, PQR is a 30° –60° –90° triangle with PQ = 2 and PR = 1. Find the value of b.
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 3

4. hypotenuse = 2 · shorter leg
Example 2
In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse.
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.
ANSWER
The length of the hypotenuse is 24. 4

5. longer leg = shorter leg · 3
Example 3
In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the longer leg.
Find Longer Leg Length
SOLUTION
The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times . 3
30° –60° –90° Triangle Theorem
longer leg = shorter leg · 3
Substitute.
= 5 · 3
ANSWER
The length of the longer leg is 3 5

6. Find the value of x. Write your answer in radical form. 1.
Now You Try 
Find Lengths in a Triangle
Find the value of x. Write your answer in radical form. 1.
ANSWER 14 2.
ANSWER 3 3.
ANSWER 10 3

7. longer leg = shorter leg · 3
Example 4
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.
Find Shorter Leg Length
SOLUTION
The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times 3 .
30° –60° –90° Triangle Theorem
longer leg = shorter leg · 3
Substitute.
5 = x · 3
= x 5 3
Divide each side by .
2.9 ≈ x
Use a calculator.
ANSWER
The length of the shorter leg is about 2.9. 7

8. hypotenuse = 2 · shorter leg Longer leg longer leg = shorter leg · 3
Example 5
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.
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.
Shorter leg
hypotenuse = 2 · shorter leg
Longer leg
longer leg = shorter leg · 3
8 = 2 · x
y = 4 · 3
4 = x
y = 4 3 8

9. The length of the shorter leg is 4.
Example 5
Find Leg Lengths
ANSWER
The length of the shorter leg is 4.
The length of the longer leg is 3 9

10. Now You 
Find Leg Lengths
Find the value of each variable. Round your answer to the nearest tenth. 4.
ANSWER
3.5 5.
ANSWER
x = 21; y = ≈ 36.4 3

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