Special Right
Triangle
Objectives:
A. Prove
the conditions for special right
triangle.
◂ Define
the two types of special right
triangle.
◂ Find
the length of the indicated side
using the two special right triangle.
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Types of Special Right
Triangle
❖ 45°-45°-90° RIGHT TRIANGLE
❖ 30°-60°-90° RIGHT TRIANGLE
Special Right Triangle
45°-45°-90° RIGHT TRIANGLE
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Special Right Triangle
45°-45°-90° RIGHT TRIANGLE
◂ is also known as an isosceles right
triangle
◂ is a right triangle with 2 equal sides or
legs (Side 1 = Side 2) and the legs are
usually labeled x.
◂ The hypotenuse is often labeled h.
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Special Right Triangle
30°-60°-90° RIGHT TRIANGLE
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Special Right Triangle
30°-60°-90° RIGHT TRIANGLE
◂ Is also known as a scalene right triangle
◂ The side across the 30° angle is the shorter
leg and often labeled s.
◂ The side across the 60° angle is the longer
leg and often labeled l.
◂ The hypotenuse is often labeled h.
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“
◂
Why are these
right triangles
special?
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Properties of 45°-45°-90° triangle
When you are finding the hypotenuse given the legs:
𝑎2 + 𝑏2 = 𝑐 2
x
h
since a = b = x, then 𝑥 2 + 𝑥 2 = ℎ2
2𝑥 2 = ℎ2
2 ∙ 𝑥 2 = ℎ2
𝒙 𝟐 = 𝒉 or
h=x 𝟐
x
In an isosceles right triangle (45-45-90 triangle), the hypotenuse is equal to the leg times 2
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Properties of 45°-45°-90° triangle
When you are finding the legs given the hypotenuse:
x
h
𝑥 2 + 𝑥 2 = ℎ2
2𝑥 2 = ℎ2
2
ℎ
𝑥2 =
2
𝑥2
x
In an isosceles right triangle (45°-45°-90°
triangle), either leg is equal to the product of
the hypotenuse and 2, divided by 2
𝑥=
ℎ2
2
=
ℎ
∙
2
2 2
𝒉 𝟐
𝒙=
𝒐𝒓
𝟐
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 ∙ 𝟐
𝒍𝒆𝒈 =
𝟐
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Properties of 30°-60°-90° triangle
∆WXZ is equilateral triangle
𝑊𝑌 is the perpendicular bisector of 𝑋𝑍
1
1
Thus, 𝑋𝑌 = 2 𝑋𝑍 = 2 𝑋𝑊, or 𝑋𝑊 = 2𝑋𝑌 = 2𝑥.
Also,
𝑋𝑌 2 + 𝑌𝑊 2 = 𝑋𝑊 2
Use Pythagorean Theorem
2
2
2
𝑠 + 𝑌𝑊 = 2𝑠
Substitute s for XY and 2s for XW
𝑌𝑊 2 = 4𝑠 2 − 𝑠 2
Substract 𝑠 2 from each side
𝑌𝑊 2 = 3𝑠 2
Simplify
𝒀𝑾 = 𝒔 𝟑
Find the square root of each side
The length of the hypotenuse is twice the length of the shorter leg. (𝒉 = 𝟐𝒔)
The length of the longer leg is 3 times the length of the shorter leg. (𝒍 = 𝒔 𝟑)
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Examples: Find the value of each
variable.
1.
Checking:
𝑐 2 = 𝑎2 + 𝑏 2
2
5 2 = 52 + 52
25 ∙ 2 = 25 + 25
𝟓𝟎 = 𝟓𝟎
ℎ=𝑥 2
𝒉=𝟓 𝟐
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Examples: Find the value of each
variable.
2.
3
x
Checking:
𝑐 2 = 𝑎2 + 𝑏 2
2
x
ℎ 2
2
𝟑 𝟐
𝒙=
𝟐
𝑥=
3 2
3 2
2
3 =
+
2
2
9×2 9×2
9=
+
4
4
9 = 4.5 + 4.5
𝟗=𝟗
2
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Examples: Find the value of each
variable.
3.
Checking:
𝑐 2 = 𝑎2 + 𝑏 2
2
𝑙=𝑠 3
𝑙
𝑠 3
=
3
3
𝑙 3
𝑠=
3
𝟕 𝟑
𝒔=
𝟑
ℎ = 2𝑠
7 3
ℎ=2
3
𝟏𝟒 𝟑
𝒉=
𝟑
2
14 3
7 3
=
+ 72
3
3
196 × 3 49 × 3
=
+ 49
9
9
588 147
=
+ 49
9
9
65.33 = 16.33 + 49
𝟔𝟓. 𝟑𝟑 = 𝟔𝟓. 𝟑𝟑
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Examples: Find the value of each
variable.
4.
Checking:
𝑐 2 = 𝑎2 + 𝑏 2
2
𝑙=𝑠 3
𝒍=𝟓 𝟑
ℎ = 2𝑠
ℎ=2 5
102 = 52 + 5 3
100 = 25 + 25 × 3
100 = 25 + 75
𝟏𝟎𝟎 = 𝟏𝟎𝟎
𝒉 = 𝟏𝟎
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Where can
we apply
the
properties of
special right
triangles.
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Solve for the following problems.
1.
An escalator lifts people to the second floor, 25 ft. above the
first floor. The escalator rises at a 30° angle. How far does a
person travel from the bottom to the top of the escalator?
Solution:
Let hypotenuse
𝒉 − be the length of the escalator
𝟐𝟓 𝒇𝒕. − be the length of the shorter leg opposite the 30° angle
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?
ℎ = 2𝑠
ℎ = 2 25 𝑓𝑡.
𝒉 = 𝟓𝟎 𝒇𝒕.
Answer:
A person travel from the top of the escalator is about 50 ft.
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2. Road Signs. The warning sign at the right is an equilateral
triangle. The height of the sign is 1m. Find the length s of each
side of the sign to the nearest tenth of a meter.
Solution:
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1
𝑙 3
𝑠=
2
3
1
1 3
𝑠=
2
3
1
𝑠=2 𝑠
2
1 3
𝑠=2
3
2 3
𝑠=
3
𝑠 = 1.155
𝒔 = 𝟏. 𝟐 𝒎𝒆𝒕𝒆𝒓
Answer:
The length 𝒔 of each side of the sign is 1.2 meter.
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Activity 1 :
A. Complete the measure of the dimensions each triangle.
3.
2
.
x = 20, y = 20 𝟑
4.
x = 24, y = 1𝟐 𝟑
B. Solve
x = 5 𝟐, y = 5
x = 12, y = 12
5. A square-shaped handkerchief measures 16 inches on each side. You
fold it along its diagonal so you can tie it around your neck. How long
is this tie? 16 𝟐 𝒊𝒏𝒄𝒉𝒆𝒔 or 22.6 inches
6. A ladder leaning against a wall makes an angle of 30 degrees with the
ground. If the length of the ladder is 9 m, find the height of the wall.
𝟗
𝒎 or 4.5 m
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ASYNCHRONOUS TASK!
◂
Proceed to your google classroom
and answer Mini Task 2 – Special
Right Triangle
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Thank you for
listening!
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