Warm-Up Exercises EXAMPLE 1 Find hypotenuse length in a 45-45-90 triang o o o Find the length of the hypotenuse. a . SOLUTION hypotenuse = leg 2 = 8 2 Substitute. 45-45-90 Triangle Theorem o o o By the Triangle Sum Theorem, the measure of the third angle must be 45 º . Then the triangle is a 45 º -45 º -90 º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg. a .
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Warm-Up Exercises EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangle o o o Find the length of the hypotenuse. a. SOLUTION hypotenuse = leg 2 = 8=
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Warm-Up ExercisesEXAMPLE 1 Find hypotenuse length in a 45-45-90 triangleo o o
Find the length of the hypotenuse.a.
SOLUTION
hypotenuse = leg 2
= 8 2 Substitute.
45-45-90 Triangle Theoremo o o
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 Theorem 7.8, the hypotenuse is 2 times as long as each leg.
a.
Warm-Up ExercisesEXAMPLE 1 Find hypotenuse length in a 45-45-90 triangleo o o
hypotenuse = leg 2
Substitute.
45-45-90 Triangle Theoremo o o
= 3 2 2= 3 2 Product of square roots
= 6 Simplify.
b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle.
45 - 45 - 90ooo
Find the length of the hypotenuse.b.
Warm-Up ExercisesEXAMPLE 2 Find leg lengths in a 45-45-90 triangleo o o
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 - 90ooo
hypotenuse = leg 2
Substitute.
45-45-90 Triangle Theoremo o o
25 = x 2
252
=2x2
5 = x
Divide each side by 2
Simplify.
Warm-Up ExercisesEXAMPLE 3 Standardized Test Practice
SOLUTION
By the Corollary to the Triangle Sum Theorem, the triangle is a triangle.45 - 45 - 90
ooo
Warm-Up ExercisesEXAMPLE 3 Standardized Test Practice
hypotenuse = leg 2
Substitute.
45-45-90 Triangle Theoremo o o
= 25 2WX
The correct answer is B.
Warm-Up ExercisesGUIDED PRACTICE for Examples 1, 2, and 3
Find the value of the variable.
1. 2. 3.
ANSWER 2 ANSWER 2 8 2ANSWER
Warm-Up ExercisesGUIDED 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 2ANSWER
Warm-Up ExercisesEXAMPLE 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°-60° triangles. The length h of the altitude is approximately the height of the logo.
h = 3 5.2 cm
3
longer leg = shorter leg 3
Warm-Up ExercisesEXAMPLE 5 Find lengths in a 30-60-90 triangleooo
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 39 = x 3
93 = x
93
33
= x
93
3 = x
3 3 = x Simplify.
Multiply fractions.
Triangle Theorem30 - 60 - 90ooo
Divide each side by 3Multiply numerator and denominator by 3
Substitute.
Warm-Up ExercisesEXAMPLE 5 Find lengths in a 30-60-90 triangleooo
hypotenuse = 2 shorter leg
STEP 2 Find the value of y.
y = 2 3 = 63 3 Substitute and simplify.
Triangle Theorem30 - 60 - 90o oo
Warm-Up ExercisesEXAMPLE 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 angleo
b. 60 angleo
SOLUTIONWhen 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
ooo
o
Warm-Up ExercisesEXAMPLE 6 Find a height
14 = h 2142
= h
9.9 h
Triangle Theorem45 - 45 - 90ooo
Divide each side by 2
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.