8.2 Special Right Triangles
Geometry
Mr. Jacob P. Gray
Franklin County High School
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8.2 Special Right Triangles
Objectives
•
•
To use the properties of 45˚-45˚-90˚ triangles
To use the properties of 30˚-60˚-90˚ triangles
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Take notes on the following:
Essential Understanding
Certain right triangles have properties that allow
you to use shortcuts to determine side lengths
without using the Pythagorean Theorem.
Right Isosceles Triangle
The acute angles of an Isosceles Right Triangle are
both 45˚. Another name for this is a 45˚-45˚-90˚
triangle.
x
x
45˚
45˚
y
𝒙𝟐 + 𝒙𝟐 = 𝒚𝟐 Use the Pythagorean Theorem
𝟐𝒙𝟐 = 𝒚𝟐 Simplify
𝒙√𝟐 = 𝒚
Take the positive square root
of each side
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Take notes on the following:
Theorem 8-5
45˚-45˚-90˚ Triangle Theorem
In a 45˚-45˚-90˚ triangle, both legs are congruent
and the length of the hypotenuse is √2 times the
length of a leg.
s√2
Hypotenuse = √2 · leg
45˚
s
45˚
s
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Here is a problem to practice.
Problem 1) Finding the Length of the Hypotenuse
What is the value of each variable?
1
2
9
45˚
x
45˚
45˚
h
Hypotenuse = √2 · leg
45˚
2√2
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Here is another problem to practice.
Problem 2) Finding the Length of a Leg
What is the value of x?
Hypotenuse = √2 · leg
6
45˚
45˚
x
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Here is another problem to practice.
Problem 3) Finding Distance
Softball A softball diamond is a square. The
distance from base to base is 60 ft. To the nearest
foot, how far does the catcher throw the ball from
home to second base?
Hypotenuse = √2 · leg
60 ft
d
𝒅 = 𝟔𝟎 · √𝟐
Use a calculator.
𝒅 ≈ 𝟖𝟒. 𝟖𝟓𝟐𝟖…
The catcher throws the ball about 85 ft from
home plate to second base.
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Another type of special right triangle is a
30˚-60˚-90˚ triangle.
Theorem 8-6
30˚-60˚-90˚ Triangle Theorem
In a 30˚-60˚-90˚ triangle, the length of the
hypotenuse is twice the length of the shorter leg.
The length of the longer leg is √3 times the length
of the shorter leg.
hypotenuse = 2 · shorter leg
longer leg = √3 · shorter leg
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30˚
2s
60˚
s
s√3
Here is a problem to practice.
Problem 4) Using the Length of One Side
Algebra What is the value of d in simplest radical
form?
5
d
60˚
30˚
f
𝒍𝒐𝒏𝒈𝒆𝒓 𝒍𝒆𝒈 = √𝟑 · 𝒔𝒉𝒐𝒓𝒕𝒆𝒓 𝒍𝒆𝒈
𝟓 = 𝒅√𝟑
𝟓
𝟑
=
𝒅 =
𝒅 𝟑
𝟑
𝟓
𝟑
∙
√𝟑
𝟓√𝟑
=
𝟑
𝟑
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Here is the last problem to practice.
Problem 5) Applying the 30˚-60˚-90˚ Triangle Theorem
Jewelry Making An artisan makes pendants in the shape
of equilateral triangles. The height of each pendant is
18mm. What is the length s of each side of a pendant to the
nearest tenth of a millimeter?
𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔 = 3 · 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔
18 = √3
1
𝑠
2
S
18mm
3
18 =
𝑠
2
2
3
∙ 18 = 𝑠
𝑠 ≈ 20.7846. .
Each side is about 20.8 mm long.
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Classwork
Pages 503-505
7-21 all
23-29 all
34, 35, 37, 38, 39
STOP
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