9.4 Special Right Triangles Geometry Mrs. Spitz Spring 2005
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9.4 Special Right Triangles
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignment
• Find the side lengths of special right triangles.
• Use special right triangles to solve real-life problems, such as finding the side lengths of the triangles.
• Assignment: pp.554-555 #1-30
• Assignment due to day: 9.3
• Quiz Monday over 9.1-9.3
Side lengths of Special Right Triangles• Right triangles whose angle
measures are 45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.
Theorem 9.8: 45°-45°-90° Triangle Theorem• In a 45°-45°-90°
triangle, the hypotenuse is √2 times as long as each leg.
x
x√2x
45°
45°
Hypotenuse = √2 ∙ leg
Theorem 9.8: 30°-60°-90° Triangle Theorem• In a 30°-60°-90°
triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.
√3x
60°
30°
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
2xx
Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
• Find the value of x• By the Triangle Sum
Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.
3 3
x
45°
Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
Hypotenuse = √2 ∙ leg
x = √2 ∙ 3
x = 3√2
3 3
x
45°
45°-45°-90° Triangle Theorem
Substitute values
Simplify
Ex. 2: Finding a leg in a 45°-45°-90° Triangle
• Find the value of x.• Because the triangle
is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg.
5
x x
Ex. 2: Finding a leg in a 45°-45°-90° Triangle
5
x x
Statement:Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
Reasons:45°-45°-90° Triangle Theorem
5
√2
√2x
√2=
5
√2x=
5
√2x=
√2
√2
5√2
2x=
Substitute values
Divide each side by √2
Simplify
Multiply numerator and denominator by √2
Simplify
Ex. 3: Finding side lengths in a 30°-60°-90° Triangle
• Find the values of s and t.
• Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
5
st
30°
60°
Ex. 3: Side lengths in a 30°-60°-90° Triangle
Statement:Longer leg = √3 ∙ shorter leg
5 = √3 ∙ s
Reasons:30°-60°-90° Triangle Theorem
5
√3
√3s
√3=
5
√3s=
5
√3s=
√3
√3
5√3
3s=
Substitute values
Divide each side by √3
Simplify
Multiply numerator and denominator by √3
Simplify
5
st
30°
60°
The length t of the hypotenuse is twice the length s of the shorter leg.
Statement:Hypotenuse = 2 ∙ shorter leg
Reasons:30°-60°-90° Triangle Theorem
t 2 ∙ 5√3
3= Substitute values
Simplify
5
st
30°
60°
t 10√3
3=
Using Special Right Triangles in Real Life• Example 4: Finding the height of a ramp.• Tipping platform. A tipping platform is a
ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?
Solution:
• When the angle of elevation is 30°, the height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet.
80 = 2h 30°-60°-90° Triangle Theorem
40 = h Divide each side by 2.
When the angle of elevation is 30°, the ramp height is about 40 feet.
Solution:
• When the angle of elevation is 45°, the height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet.
80 = √2 ∙ h 45°-45°-90° Triangle Theorem
80
√2= h Divide each side by √2
Use a calculator to approximate56.6 ≈ hWhen the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.
Ex. 5: Finding the area of a sign• Road sign. The
road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle.
h
18 in.
36 in.
Ex. 5: Solution
• First, find the height h of the triangle by dividing it into two 30°-60°-90° triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 18 inches.
h = √3 ∙ 18 = 18√330°-60°-90° Triangle
Theorem
h
18 in.
36 in.
Use h = 18√3 to find the area of the equilateral triangle.
Ex. 5: Solution
Area = ½ bh = ½ (36)
(18√3) ≈ 561.18
h
18 in.
36 in.
The area of the sign is a bout 561 square inches.
Reminders:
• Wednesday, March 1 – Test Corrections
• Friday, March 3 – Quiz over 9.1-9.3
• Wednesday, March 8– Quiz over 9.4-9.5
• March 8 ALL WORK MUST BE TURNED IN!!!!!
• Monday, March 13 – Chapter 9 Test/
• Binder Check – Same day
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