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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
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45-45-90 Triangle Theorem
Substitute.
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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
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the Triangle Sum Theorem, the triangle is a 45triangle.
- 45 - 90
hypotenuse = leg
=3 2
=3 2
=6
2
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45-45-90 Triangle Theorem
2 Substitute.
Product of square roots
Simplify.
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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
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Triangle Sum Theorem, the triangle is a 45 - 45 - 90
triangle.
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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
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triangle is a 45 - 45 - 90 triangle.
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
hypotenuse = leg
WX = 25
2
2
The correct answer is B.
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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
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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
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30 - 60 - 90 Triangle Theorem
Substitute.
Divide each side by
3
Multiply numerator and
denominator by 3
Multiply fractions.
Simplify.
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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
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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.
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45 angle
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b. 60 angle
SOLUTION
a. When the body is raised 45oabove the frame, the
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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
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45 - 45 - 90 Triangle Theorem
Divide each side by
2
Use a calculator to approximate.
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When the angle of elevation is 45, the body is about
9 feet 11 inches above the frame.
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b. When the body is raised 60, the height h is the
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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
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30 - 60 - 90 Triangle Theorem
Substitute.
Use a calculator to approximate.
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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