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Special Right Triangles 5.1 (M2)
Special Right Triangles
5.1 (M2)
Pythagorean Theorem
Right Triangle Theorems
45o-45o-90o Triangle Theorem Hypotenuse is times as long as each
leg
30o-60o-90o Triangle Theorem Hypotenuse is twice as long as the
shorter leg, and the longer leg is times as long as the shorter leg
2
3
EXAMPLE 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.
EXAMPLE 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.
EXAMPLE 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 = xDivide each side by 2Simplify.
EXAMPLE 3 Standardized Test Practice
SOLUTION
By the Corollary to the Triangle Sum Theorem, the triangle is a triangle.45 - 45 - 90
ooo
EXAMPLE 3 Standardized Test Practice
hypotenuse = leg 2
Substitute.
45-45-90 Triangle Theoremo o o
= 25 2WX
The correct answer is B.
GUIDED PRACTICE for Examples 1, 2, and 3
Find the value of the variable.
1. 2. 3.
ANSWER 2 ANSWER 2 8 2ANSWER
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.
3 2ANSWER
EXAMPLE 4 Find the height of an equilateral triangle
LogoThe logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo?
SOLUTIONDraw 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
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 3Multiply numerator and denominator by 3
Substitute.
EXAMPLE 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
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 angle
ob. 60 angle
o
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
oooo
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
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
GUIDED PRACTICE for Examples 4, 5, and 6
Find the value of the variable.
ANSWER 3 ANSWER 3 2
GUIDED 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.
