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o o o Warm-Up1Exercises EXAMPLE Find hypotenuse length in a 45-45-90 triangle Find the length of the hypotenuse. a. SOLUTION a. By the Triangle Sum Theorem, the measure of the third angle must be 45º. Then the triangle is a 45º-45º90º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg. hypotenuse = leg =8 2 2 o o o 45-45-90 Triangle Theorem Substitute. o o o Warm-Up1Exercises EXAMPLE Find hypotenuse length in a 45-45-90 triangle Find the length of the hypotenuse. b. b. By the Base Angles Theorem and the Corollary to o o o the Triangle Sum Theorem, the triangle is a 45triangle. - 45 - 90 hypotenuse = leg =3 2 =3 2 =6 2 o o o 45-45-90 Triangle Theorem 2 Substitute. Product of square roots Simplify. o o o Warm-Up Exercises EXAMPLE 2 Find leg lengths in a 45-45-90 triangle Find the lengths of the legs in the triangle. SOLUTION By the Base Angles Theorem and the Corollaryoto the o o Triangle Sum Theorem, the triangle is a 45 - 45 - 90 triangle. o o o 45-45-90 Triangle Theorem hypotenuse = leg 2 5 2 =x 2 5 x 2 2 = 2 2 5=x Substitute. Divide each side by Simplify. 2 Warm-Up3Exercises EXAMPLE Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the o o o triangle is a 45 - 45 - 90 triangle. Warm-Up3Exercises EXAMPLE Standardized Test Practice hypotenuse = leg WX = 25 2 2 The correct answer is B. o o o 45-45-90 Triangle Theorem Substitute. Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2, and 3 Find the value of the variable. 1. ANSWER 2. 2 3. ANSWER 2 ANSWER 8 2 Warm-Up Exercises 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. ANSWER 3 2 Warm-Up4Exercises EXAMPLE 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°-60° triangles. The length h of the altitude is approximately the height of the logo. longer leg = shorter leg h=3 3 3 5.2 cm o o o Warm-Up5Exercises EXAMPLE Find lengths in a 30-60-90 triangle Find the values of x and y. Write your answer in simplest radical form. STEP 1 Find the value of x. longer leg = shorter leg 9=x 3 9 =x 3 9 3 =x 3 3 9 3 =x 3 3 3 =x 3 o o o 30 - 60 - 90 Triangle Theorem Substitute. Divide each side by 3 Multiply numerator and denominator by 3 Multiply fractions. Simplify. o o o Warm-Up5Exercises EXAMPLE Find lengths in a 30-60-90 triangle STEP 2 Find the value of y. hypotenuse = 2 shorter leg y=2 3 3 =6 3 o o o 30 - 60 - 90 Triangle Theorem Substitute and simplify. Warm-Up6Exercises EXAMPLE 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. o 45 angle o b. 60 angle SOLUTION a. When the body is raised 45oabove the frame, the o o o height h is the length of a leg of a 45 - 45 - 90 triangle. The length of the hypotenuse is 14 feet. Warm-Up6Exercises EXAMPLE Find a height 14 = h 2 14 =h 2 9.9 h o o o 45 - 45 - 90 Triangle Theorem Divide each side by 2 Use a calculator to approximate. o 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 o o o length of the longer leg of a 30 - 60 - 90 triangle. The length of the hypotenuse is 14 feet. Warm-Up6Exercises EXAMPLE Find a height hypotenuse = 2 shorter leg 30o- 60o- 90o Triangle Theorem 14 = 2 s Substitute. 7 =s Divide each side by 2. longer leg = shorter leg h =7 3 h 12.1 3 o o o 30 - 60 - 90 Triangle Theorem Substitute. Use a calculator to approximate. o When the angle of elevation is 60, the body is about 12 feet 1 inch above the frame. Warm-Up Exercises GUIDED PRACTICE for Examples 4, 5, and 6 Find the value of the variable. ANSWER 3 ANSWER 2 3 Warm-Up Exercises GUIDED PRACTICE for Examples 4, 5, and 6 7. What If? In Example 6, what is the height of the body of the dump truck if it is raised 30° above the frame? ANSWER 8. 7 ft In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side? SAMPLE ANSWER The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle. Daily Homework Quiz Warm-Up Exercises Use these triangles for exercises 1- 4. 1. Find a if b = 10 2 ANSWER 2. 10 Find b if a = 19 ANSWER 19 2 Daily Homework Quiz Warm-Up Exercises Use these triangles for exercises 1- 4. 3. Find d and e if c = 4. ANSWER 4. d=4 3,e=8 Find c and d if e = 50 3 . ANSWER c = 25 3 , d = 75 Daily Homework Quiz Warm-Up Exercises 5. Find x, y and z. ANSWER x= 3 2 ,y= 3 6 ,z= 6 2