example 4 find the height of an equilateral triangle logo the logo on the recycling bin at the right...
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EXAMPLE 4 Find the height of an equilateral triangle
Logo
The logo on the recycling bin at the right resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo?
SOLUTION
30 - 60 - 90ooo
Draw the equilateral triangle described. Its altitude forms the longer leg of two triangles. The length h of the altitude is approximately the height of the logo.
EXAMPLE 4 Find the height of an equilateral triangle
longer leg = shorter leg 3
h = 3 5.2 cm
3
EXAMPLE 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 3
Multiply numerator and denominator by 3
EXAMPLE 5 Find lengths in a 30-60-90 triangleooo
longer leg = 2 shorter leg
STEP 2Find the value of y.
y = 2 3 = 63 3 Substitute and simplify.
Triangle Theorem30 - 60 - 90ooo
EXAMPLE 6 Find a height
Dump TruckThe 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
EXAMPLE 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
EXAMPLE 6 Find a height
longer leg = 2 shorter leg Triangle Theorem30 - 60 - 90ooo
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
GUIDED PRACTICE for Examples 4, 5 and 6
Find the value of the variable.
SOLUTION
longer leg = shorter leg 3 Triangle Theorem30 - 60 - 90ooo
Substitute.
Simplify.
x = 3 3
x 3=
GUIDED PRACTICE for Examples 4, 5 and 6
Find the value of the variable.
All side are equal, therefore it is an equilateral triangle a 30° - 60° - 90° triangle can be found by dividing an equilateral triangle in half
longer leg = 2 shorter leg Triangle Theorem30 - 60 - 90ooo
SOLUTION
Substitute.
Simplify.
h = 2 3h = 3 2
GUIDED PRACTICE for Examples 4, 5 and 6
Find the value of the variable.
SOLUTION
Triangle Theorem30 - 60 - 90oooHypotenuse = 2 shorter leg
Substitute.
Divide both sides by 214 2
= x
7 = x
14 2 x=
Simplify.
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.
GUIDED PRACTICE for Examples 4, 5 and 6
Find the value of the variable.
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.