7.4 special right triangles

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

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 .

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.


Example 1

.

.

.

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.


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


The correct answer is B.ANSWER

7.4 Guided Practice

Find the value of the variable.

ANSWER 2

1.

7.4


ANSWER 2

2.

Guided Practice

7.4


8 2ANSWER

3.

Guided Practice

7.4

4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of 6.

3 2ANSWER

Guided Practice

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

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


Multiply numerator and denominator by 3

Substitute.

7.4

hypotenuse = 2 shorter leg

STEP 2 Find the value of y.

y = 2 3 = 63 3 Substitute and simplify.


Example 5

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.

7.4

14 = h 2142

= h

9.9 h


Use a calculator to approximate.

When the angle of elevation is 45°, the body is about 9 feet 11 inches above the frame.


Example 6

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.



Example 6

7.4 Guided Practice


ANSWER 3

5.

7.4


ANSWER 3 2

6.

Guided Practice

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

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

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

7.4

5. Find x, y and z.

ANSWER 3 2x = 6 2z = 3 6y = , ,

Exit Slip

7.4

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7.4 special right triangles

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