8.2-2 Special
Right Triangles
Geometry
Mr. Peebles
Spring 2013
Bell Ringer : Find The Value of x.
10
x
x
Bell Ringer : Find The Value of x.
10
x
Answer:
x5 2
x
Daily Learning Target (DLT)
• “I can apply the properties of 30-60-90
right triangles in mathematical and realworld problems.”
Assignment: Due Now
• Pgs. 428-429 (1-8, 29)
Theorem 9.8: 30°-60°-90°
Triangle Theorem
• In a 30°-60°-90°
triangle, the
hypotenuse is
twice as long as x
the shorter leg,
and the longer leg
is √3 times as long
as the shorter leg.
60°
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
2x
30°
√3x
Theorem 9.8: 30°-60°-90°
Triangle Theorem – In Other
Words
Angle
Opposite Side
30 Degrees
X
60 Degrees
3x
90 Degrees
2x
60°
2x
x
30°
√3x
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°
t
30°
60°-90° triangle,
5
the longer leg is
√3 times the
length s of the
Hypotenuse = 2 ∙ shorter leg
shorter leg.
Longer leg = √3 ∙ shorter leg
s
Ex. 3: Side lengths in a 30°-60°-90°
Triangle
t
60°
s
30°
5
Statement:
Reasons:
Longer leg = √3 ∙ shorter leg
5 = √3 ∙ s
5
√3
5
√3
√3
√3
5
√3
5√3
3
=
√3s
√3
= s
30°-60°-90° Triangle Theorem
Substitute values
Divide each side by √3
Simplify
= s
Multiply numerator and
denominator by √3
= s
Simplify
The length t of the hypotenuse is twice the length s of the
shorter leg.
60°
t
s
30°
5
Statement:
Reasons:
Hypotenuse = 2 ∙ shorter leg
5√3
= 2∙
3
t
t
=
10√3
3
30°-60°-90° Triangle Theorem
Substitute values
Simplify
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?
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
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.
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.
18 in.
h
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√3
30°-60°-90° Triangle
Theorem
18 in.
h
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
18 in.
h
36 in.
The area of the sign is
a bout 561 square
inches.
Assignment:
• Pgs. 428-429 (9-15, 17-19, 23, 25,
30, 31)-Due Today
Exit Quiz: Find The Value of x.
(5 Points)
12
x
x
Exit Quiz – 5 Points
• Find the values of
s and t.
• Because the
triangle is a 30°-
60°
t
30°
60°-90° triangle,
5
the longer leg is
√3 times the
length s of the
Hypotenuse = 2 ∙ shorter leg
shorter leg.
Longer leg = √3 ∙ shorter leg
s