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

4. Find m DBC in square ABCD. ANSWER 45

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. 45-45-90 Triangle Theorem o √2 hypotenuse = leg √2 = 8 Substitute.

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. 45 - 45 - 90 o 45-45-90 Triangle Theorem o hypotenuse = leg 2 = 3 2 Substitute. = 3 2 Product of square roots = 6 Simplify.

Find leg lengths in a 45-45-90 triangle EXAMPLE 2 o 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. 45 - 45 - 90 o 45-45-90 Triangle Theorem o hypotenuse = leg 2 2 5 = x Substitute. 2 5 = x Divide each side by 2 5 = x Simplify.

Standardized Test Practice EXAMPLE 3 Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the triangle is a triangle. 45 - 45 - 90 o

Standardized Test Practice EXAMPLE 3 Standardized Test Practice o o o hypotenuse = leg 2 45-45-90 Triangle Theorem = 25 2 WX Substitute. The correct answer is B.

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

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

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 30° −60° −90° triangles. The length h of the altitude is approximately the height of the logo. longer leg = shorter leg 3 h = 3 5.2 cm 3

Find lengths in a 30-60-90 triangle EXAMPLE 5 o 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 30 - 60 - 90 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

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

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. 45 - 45 - 90 o

EXAMPLE 6 Find a height 45 - 45 - 90 14 = h 2 14 2 = h 9.9 h o 14 = h 2 Triangle Theorem 14 2 = h Divide each side by 2 9.9 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. 30 - 60 - 90 o

hypotenuse = 2 shorter leg EXAMPLE 6 Find a height Triangle Theorem 30 - 60 - 90 o hypotenuse = 2 shorter leg 14 = 2 s Substitute. 7 = s Divide each side by 2. Triangle Theorem 30 - 60 - 90 o longer leg = shorter leg 3 h = 7 3 Substitute. h 12.1 Use a calculator to approximate. When the angle of elevation is 60, the body is about 12 feet 1 inch above the frame. o

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

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.

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

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

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