1. Simplify.
ANSWER 12 2. 6 3
ANSWER
2 3 2
ANSWER
5 2 2

2. 4. Find m DBC in square ABCD.
ANSWER 45

3. Find hypotenuse length in a 45-45-90 triangle EXAMPLE 1
Find the length of the hypotenuse. a.
SOLUTION
By the Triangle Sum Theorem, the measure of the third angle must be 45º. Then the triangle is a 45º −45º −90º triangle, so by the 45º −45º −90º Triangle Theorem, the hypotenuse is √2 times as long as each leg. a.
Triangle Theorem o √2
hypotenuse = leg √2
= 8
Substitute.

4. Find hypotenuse length in a 45-45-90 triangle EXAMPLE 1
Find the length of the hypotenuse. b.
SOLUTION b.
By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. o
Triangle Theorem o
hypotenuse = leg 2
= 3 2
Substitute.
= 3 2
Product of square roots
= 6
Simplify.

5. Find leg lengths in a 45-45-90 triangle EXAMPLE 2o
EXAMPLE 2
Find the lengths of the legs in the triangle.
SOLUTION
By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle. o
Triangle Theorem o
hypotenuse = leg 2 2 5
= x
Substitute. 2 5 = x
Divide each side by 2
5 = x
Simplify.

6. Standardized Test Practice
EXAMPLE 3
Standardized Test Practice
SOLUTION
By the Corollary to the Triangle Sum Theorem, the triangle is a triangle. o

7. Standardized Test Practice
EXAMPLE 3
Standardized Test Practice o o o
hypotenuse = leg 2
Triangle Theorem
= 25 2 WX
Substitute.
The correct answer is B.

8. GUIDED PRACTICE
for Examples 1, 2, and 3
Find the value of the variable. 3. 1. 2.
ANSWER 2
ANSWER 2 8 2
ANSWER

9. GUIDED PRACTICE
for Examples 1, 2, and 3
4. Find the leg length of a 45°- 45°- 90° triangle with a
hypotenuse length of 6. 3 2
ANSWER

10. EXAMPLE 4
Find the height of an equilateral triangle
Logo
The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo?
SOLUTION
Draw the equilateral triangle described. Its altitude forms the longer leg of two ° −60° −90° triangles. The length h of the altitude is approximately the height of the logo.
longer leg = shorter leg 3
h = cm 3

11. Find lengths in a 30-60-90 triangle EXAMPLE 5o
EXAMPLE 5
Find the values of x and y. Write your answer in simplest radical form.
STEP 1
Find the value of x.
Triangle Theorem o
longer leg = shorter leg 3
9 = x 3
Substitute. 9 3
= x
Divide each side by 3
Multiply numerator and denominator by 3 9 3
= x 9 3
= x
Multiply fractions.
Simplify. 3
= x

12. Find lengths in a 30-60-90 triangle EXAMPLE 5o
EXAMPLE 5
STEP 2
Find the value of y.
Triangle Theorem o
hypotenuse = 2 shorter leg
y = = 6 3
Substitute and simplify.

13. EXAMPLE 6 Find a height Dump Truck
The body of a dump truck is raised to empty a load of sand. How high is the 14 foot body from the frame when it is tipped upward at the given angle? a.
45 angle o b.
60 angle o
SOLUTION
When the body is raised 45 above the frame, the height h is the length of a leg of a triangle. The length of the hypotenuse is 14 feet. a. o

14. EXAMPLE 6 Find a height 45 - 45 - 90 14 = h 2 14 2 = h 9.9 ho
14 = h 2
Triangle Theorem 14 2
= h
Divide each side by 2 h
Use a calculator to approximate.
When the angle of elevation is 45, the body is about 9 feet 11 inches above the frame. o b.
When the body is raised 60, the height h is the length of the longer leg of a triangle. The length of the hypotenuse is 14 feet. o

15. hypotenuse = 2 shorter leg
EXAMPLE 6
Find a height
Triangle Theorem o
hypotenuse = 2 shorter leg
14 = 2 s
Substitute.
7 = s
Divide each side by 2.
Triangle Theorem o
longer leg = shorter leg 3
h = 7 3
Substitute. h
Use a calculator to approximate.
When the angle of elevation is 60, the body is about feet 1 inch above the frame. o

16. Find the value of the variable.
GUIDED PRACTICE
for Examples 4, 5, and 6
Find the value of the variable.
ANSWER 3
ANSWER 3 2

17. GUIDED PRACTICE
for Examples 4, 5, and 6
WHAT IF? In Example 6, what is the height of the body of the dump truck if it is raised 30° above the frame? 7.
ANSWER
7 ft
In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side. 8.
SAMPLE ANSWER
The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle.

18. Daily Homework Quiz
Use these triangles for exercises 1- 4. 1.
Find a if b = 2 10
ANSWER 10 2.
Find b if a = 19
ANSWER 2 19

19. Daily Homework Quiz
Use these triangles for exercises 1- 4. 3.
Find d and e if c = 4.
ANSWER
d = 4 3
, e = 8 4. 3 50
Find c and d if e = .
ANSWER 3 25
c =
, d = 75

20. Daily Homework Quiz 5.
Find x, y and z.
ANSWER 3 2
x = 6
z =
y = ,