7.4 special right triangles
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7.47.4 Special Right TrianglesBell Thinger
Simplify.
1. 6 2 2
2.36
ANSWER 12
ANSWER 2 3
23. 5
ANSWER 5 22
4. Find m DBC in square ABCD.
ANSWER 45
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7.4
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7.4Example 1
Find the length of the hypotenuse.a.
SOLUTION
2= 8 Substitute.
45°- 45°- 90° Triangle Theorem
a. 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.
2hypotenuse = leg .
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7.4
Find the length of the hypotenuse.
SOLUTION
b.
2hypotenuse = leg
Substitute.22= 3
= 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 45º- 45º- 90º triangle.
45°- 45°- 90° Triangle Theorem
Example 1
.
.
.
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7.4 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 45º- 45º- 90º triangle.
hypotenuse = leg 2
Substitute.25 = x 2
252
=2x2
5 = x
Divide each side by 2
Simplify.
45°- 45°- 90° Triangle Theorem
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7.4 Example 3
SOLUTION
By the Corollary to the Triangle Sum Theorem, the triangle is a 45º- 45º- 90º triangle.
hypotenuse = leg 2
Substitute.= 25 2WX
45°- 45°- 90° Triangle Theorem
The correct answer is B.ANSWER
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7.4 Guided Practice
Find the value of the variable.
ANSWER 2
1.
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7.4
Find the value of the variable.
ANSWER 2
2.
Guided Practice
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7.4
Find the value of the variable.
8 2ANSWER
3.
Guided Practice
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7.4
4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6.
3 2ANSWER
Guided Practice
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7.4
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7.4 Example 4
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
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7.4 Example 5
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.
30° - 60° - 90° Triangle Theorem
Divide each side by 3
Multiply numerator and denominator by 3
Substitute.
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7.4
hypotenuse = 2 shorter leg
STEP 2 Find the value of y.
y = 2 3 = 63 3 Substitute and simplify.
30° - 60° - 90° Triangle Theorem
Example 5
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7.4 Example 6
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
SOLUTION
b. 60° angle
a. When the body is raised 45 above the frame, the height h is the length of a leg of a 45°- 45°- 90° triangle. The length of the hypotenuse is 14 feet.
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7.4
14 = h 2142
= h
9.9 h
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.
45° - 45° - 90° Triangle Theorem
Example 6
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7.4
b. When the body is raised 60°, the height h is the length of the longer leg of a 45°- 45°- 90° triangle. The length of the hypotenuse is 14 feet.
hypotenuse = 2 shorter leg
14 = 2 s Substitute.
7 = s Divide each side by 2.
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.
30° - 60° - 90° Triangle Theorem
30° - 60° - 90° Triangle Theorem
Example 6
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7.4 Guided Practice
Find the value of the variable.
ANSWER 3
5.
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7.4
Find the value of the variable.
ANSWER 3 2
6.
Guided Practice
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7.4
SAMPLE ANSWER
The shorter side is adjacent to the 60° angle, the longer side is adjacent to the 30° angle.
8. In a 30°- 60°- 90° triangle, describe the location of the shorter side. Describe the location of the longer side?
Guided Practice
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7.4Exit Slip
Use these triangles for Exercises 1- 4.
1. Find a if b = 10 2
ANSWER 10
2. Find b if a = 19
ANSWER 19 2
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7.4
Use these triangles for Exercises 1- 4.
3. Find d and e if c = 4.
ANSWER d = 4 3 , e = 84. 50 3Find c and d if e = .
ANSWER 25 3c = , d = 75
Exit Slip
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7.4
5. Find x, y and z.
ANSWER 3 2x = 6 2z = 3 6y = , ,
Exit Slip
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7.4
HomeworkPg #