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Lesson 8-2
Special Right Triangles
Lesson Objectives
1 Use the properties of 45-45-90
triangles
2 Use the properties of 30-60-90
triangles
NAEP 2005 Strand: Geometry
Topic: Relationships Among Geometric Figures
Local Standards: ____________________________________
Key Concepts.
In a 45-45-90 triangle, both legs are
length of the hypotenuse is
hypotenuse and the
s2
45 s
times the length of a leg.
45
s
?
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Theorem 8-5: 45-45-90 Triangle Theorem
Theorem 8-6: 30-60-90 Triangle Theorem
In a 30-60-90 triangle, the length of the hypotenuse is
is
. The length of the longer leg
times the length of the
hypotenuse longer leg 2s
.
?
30 s3
60
s
?
Examples.
1 Finding the Length of the Hypotenuse Find the value of the
variable. Use the 45-45-90 Triangle Theorem to find the
hypotenuse.
h
? 5!6
h
hypotenuse 45
h
? leg
5 6
Simplify.
45
( )
h 5(
)
h5
5 6
h
The length of the hypotenuse is
146
Geometry Lesson 8-2
.
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the length of the
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2 Applying the 45-45-90 Triangle Theorem The distance from
one corner to the opposite corner of a square playground is 96 ft.
To the nearest foot, how long is each side of the playground?
The distance from one corner to the opposite corner, 96 ft, is the
length of the hypotenuse of a 45-45-90 triangle.
s
96 s
leg Divide each side by
leg Use a calculator.
? leg
.
Each side of the playground is about
ft.
3 Using the Length of One Side Find the value of each variable.
Use the 30-60-90 Triangle Theorem to find the lengths of the legs.
4!3 © Pearson Education, Inc., publishing as Pearson Prentice Hall.
hypotenuse ? leg
96 ft
hypotenuse ?x
4 3
x
Divide each side by
x
Simplify.
y
.
60
y
? shorter leg
y
? 2!3
y2?
30
? shorter leg
?
x
Triangle Theorem
Substitute
for shorter leg.
Simplify.
y
The length of the shorter leg is
, and the length of the longer leg is
.
Quick Check.
1. Find the length of the hypotenuse of a 45-45-90 triangle with legs of length 5 !3.
2. A square garden has sides of 100 ft long. You want to build a brick path along
a diagonal of the square. How long will the path be? Round your answer to the
nearest foot.
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Geometry Lesson 8-2
147
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Example.
4 Applying the 30-60-90 Triangle Theorem A rhombus-shaped
ft
garden has a perimeter of 100 ft and a 60 angle. Find the area of the
garden to the nearest square foot.
Because a rhombus has four sides of equal length, each side is
ft.
60°
Because a rhombus is also a parallelogram, you can use the area formula
A
. Draw the altitude h, and then solve for h.
? shorter leg
shorter leg hypotenuse ? shorter leg
Divide each side by
.
h 12.5
longer leg A bh
Use the formula for the area of a
A
(
)(12.5!3)
A
To the nearest square foot, the area is
? shorter leg
Substitute
for b and
.
for h.
Use a calculator.
ft 2.
Quick Check.
3. Find the value of each variable.
4. A rhombus has 10-in. sides, two of which meet to form the indicated angle.
Find the area of each rhombus. (Hint: Use a special right triangle to find height.)
a. a 30 angle
b. a 60 angle
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Geometry Lesson 8-2
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25 All rights reserved.
The height h is the longer leg of the right triangle. To find the height h, you
can use the properties of 30-60-90 triangles. Then apply the
area formula.