Warm-Up Exercises EXAMPLE 1 Find hypotenuse length in a 45-45-90 triangle o o o Find the length of...
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![Page 1: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/1.jpg)
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.
![Page 2: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/2.jpg)
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.
![Page 3: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/3.jpg)
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.
![Page 4: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/4.jpg)
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
![Page 5: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/5.jpg)
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.
![Page 6: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/6.jpg)
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
![Page 7: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/7.jpg)
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
![Page 8: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/8.jpg)
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
![Page 9: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/9.jpg)
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.
![Page 10: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/10.jpg)
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
![Page 11: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/11.jpg)
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
![Page 12: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/12.jpg)
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.
30 - 60 - 90ooo
o
![Page 13: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/13.jpg)
Warm-Up ExercisesEXAMPLE 6 Find a height
hypotenuse = 2 shorter leg Triangle Theorem30 - 60 - 90o oo
14 = 2 s Substitute.
7 = s Divide each side by 2.
longer leg = shorter leg 3 Triangle Theorem30 - 60 - 90ooo
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
![Page 14: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/14.jpg)
Warm-Up ExercisesGUIDED PRACTICE for Examples 4, 5, and 6
Find the value of the variable.
ANSWER 3 ANSWER 3 2
![Page 15: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/15.jpg)
Warm-Up ExercisesGUIDED 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
SAMPLE ANSWER
The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle.
In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side?
8.
![Page 16: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/16.jpg)
Warm-Up ExercisesDaily Homework Quiz
Use these triangles for exercises 1- 4.
1. Find a if b = 210
ANSWER 10
2. Find b if a = 19
ANSWER 219
![Page 17: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/17.jpg)
Warm-Up ExercisesDaily Homework Quiz
Use these triangles for exercises 1- 4.
3. Find d and e if c = 4.
ANSWER d = 4 3 , e = 8
4. 350Find c and d if e = .
ANSWER 325c = , d = 75
![Page 18: 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=](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bfbd1a28abf838ca1f4e/html5/thumbnails/18.jpg)
Warm-Up ExercisesDaily Homework Quiz
5. Find x, y and z.
ANSWER 3 2x = 6 2z = 3 6y = , ,