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Geometry
9.4 Special Right Triangles
April 21, 2023 Geometry 9.4 Special Right Triangles 2
Goals
I will know the side lengths of special right triangles.
I can use special right triangles to solve problems.
April 21, 2023 Geometry 9.4 Special Right Triangles 3
First, radical review:
A radical in simplest form has no perfect squares in the radicand.
20 4 5 4 5 2 5
150 25 6 25 6 5 6
45 9 5 3 5
April 21, 2023 Geometry 9.4 Special Right Triangles 4
Rationalizing Denominators
This means no square roots allowed in the denominator of fractions.
General rule is to multiply numerator and denominator by the radical and simplify.
a a n a n
nn n n
April 21, 2023 Geometry 9.4 Special Right Triangles 5
Examples
10 10 2 10 25 2
22 2 2
24 24 3 24 38 3
33 3 3
9 9 2 9 24.5 2
22 2 2
April 21, 2023 Geometry 9.4 Special Right Triangles 6
April 21, 2023 Geometry 9.4 Special Right Triangles 7
Constructing Special Triangles
Construct an equilateral triangle.
Each angle measures ______.60
60 60
60
April 21, 2023 Geometry 9.4 Special Right Triangles 8
Constructing Special Triangles
Draw an altitude. It is perpendicular
to the base. It also bisects the
vertex angle.60 60
6030
April 21, 2023 Geometry 9.4 Special Right Triangles 9
Constructing Special Triangles
Clean up the drawing – only keep the triangle on the left.
60 60
30
April 21, 2023 Geometry 9.4 Special Right Triangles 10
Constructing Special Triangles
This is called a 30 – 60 – 90 triangle.
60
30
April 21, 2023 Geometry 9.4 Special Right Triangles 11
Constructing Special Triangles
Give the original side an arbitrary length. Call it 2a.
60
30
2a
April 21, 2023 Geometry 9.4 Special Right Triangles 12
Constructing Special Triangles
What is the length of the base?
60
30
2a
2a?a
April 21, 2023 Geometry 9.4 Special Right Triangles 13
Constructing Special Triangles
Now find the height of the triangle, h.
a2 + h2 = (2a)2
60
30
2a
a
h
April 21, 2023 Geometry 9.4 Special Right Triangles 14
Constructing Special Triangles
2 2 2
2 2 2
2 2
2
(2 )
4
3
3
3
a h a
a h a
h a
h a
h a
60
30
2a
a
h 3a
April 21, 2023 Geometry 9.4 Special Right Triangles 15
30-60-90 Triangle
60
30
2a
a
3a
April 21, 2023 Geometry 9.4 Special Right Triangles 16
Example 1
Find x & y.
60
30
2a
a
3a
60
30
12
x
y
6
April 21, 2023 Geometry 9.4 Special Right Triangles 17
Example 2
Find x & y.
60
30
2a
a
3a
60
30
8
x y16
April 21, 2023 Geometry 9.4 Special Right Triangles 18
Example 3
Find x & y.
60
30
2a
a
3a
60
30
40
x
y
20
April 21, 2023 Geometry 9.4 Special Right Triangles 19
Example 4
Find x & y.
60
30
2a
a
3a
60
30
x
y 12
April 21, 2023 Geometry 9.4 Special Right Triangles 20
Example 4 Solution3 12
12
3
12 3
3 3
12 3
3
4 3
x
x
x
x
x
4 360
30
x
y 128 3
April 21, 2023 Geometry 9.4 Special Right Triangles 21
30-60-90 Triangle
60
30
2a
a
3a
Learn the pattern!
April 21, 2023 Geometry 9.4 Special Right Triangles 22
Try it. Find x and y.
60
30
x
15
y30
April 21, 2023 Geometry 9.4 Special Right Triangles 23
Another one: Find h and y.
60
30
y
h14
7
April 21, 2023 Geometry 9.4 Special Right Triangles 24
Again: Find x and y.
60
30
y
x
2
4
April 21, 2023 Geometry 9.4 Special Right Triangles 25
Drawing Special Triangles
Draw a Square. Draw one
diagonal. The diagonal
bisects the angles. 1 & 2 measure
______.45 1
2
April 21, 2023 Geometry 9.4 Special Right Triangles 26
Drawing Special Triangles
Clean it up – keep only the triangle on the right.
45
45
April 21, 2023 Geometry 9.4 Special Right Triangles 27
Constructing Special Triangles
In the original square, each side was the same. Label them a.
45
45
a
a
April 21, 2023 Geometry 9.4 Special Right Triangles 28
Constructing Special Triangles
Solve for the hypotenuse, c.
45
45
a
ac2 2 2
2 2
2
2
2
2
c a a
c a
c a
c a
April 21, 2023 Geometry 9.4 Special Right Triangles 29
45-45-90 Triangle
45
45
a
a
April 21, 2023 Geometry 9.4 Special Right Triangles 30
Example 5
Solve for x & y.
45
45
5
xya
45
45a
a25
April 21, 2023 Geometry 9.4 Special Right Triangles 31
Example 6
Solve for x & y.
45
45
8
x
y
a45
45a
a2 8
April 21, 2023 Geometry 9.4 Special Right Triangles 32
Example 7: A tougher one.
Solve for x & y.
45
45
x10
y
a45
45a
a2
April 21, 2023 Geometry 9.4 Special Right Triangles 33
Example 7 Solution
45
45
x10
y
2 10
10
2
10 2
2 2
10 25 2
2
a
a
April 21, 2023 Geometry 9.4 Special Right Triangles 34
45-45-90 Triangle
45
45
a
a
Learn the pattern!
April 21, 2023 Geometry 9.4 Special Right Triangles 35
Try it. Find x and y.
45
45
x
y8
April 21, 2023 Geometry 9.4 Special Right Triangles 36
45-45-90 Triangle
12 2a
45
45
a
Shortcut pattern:
12 2a
April 21, 2023 Geometry 9.4 Special Right Triangles 37
Example 8 Solve for x & y
40
x
y
2 2 2
2
2
40
2 1600
800
800
400 2
20 2
x x
x
x
x
x
x
April 21, 2023 Geometry 9.4 Special Right Triangles 38
Example 9
A designer wants to put a rope light on the diagonal of a square dance floor. If the floor measures 30 ft. on a side, how long does the rope light need to be?
April 21, 2023 Geometry 9.4 Special Right Triangles 39
Example 9 Solution
30 ft.
30 ft.
45
45
April 21, 2023 Geometry 9.4 Special Right Triangles 40
Summary
60
30
2a
a
3a
45
45
a
a
45-45-90 30-60-90In Trigonometry, you will understand why these triangle are so important.
April 21, 2023 Geometry 9.4 Special Right Triangles 41
Quick Quiz – 5 problems
30
xy
12
1. Find x & y.
24
April 21, 2023 Geometry 9.4 Special Right Triangles 42
60
x
y
50
2. Find x & y.
25
April 21, 2023 Geometry 9.4 Special Right Triangles 43
75
60
x
3. Find the area of the triangle.
3 75
75 3
3 3
25 3
x
x
x
April 21, 2023 Geometry 9.4 Special Right Triangles 44
75
60
253
3. Find the area of the triangle.
175 25 3
21623.8
A
A
April 21, 2023 Geometry 9.4 Special Right Triangles 45
4. Find x.
62x
2 62
62 2
2 2
62 231 2
2
x
x
x
45
April 21, 2023 Geometry 9.4 Special Right Triangles 46
5. Find the area.
45
7
7
17 7 24.5
2A
April 21, 2023 Geometry 9.4 Special Right Triangles 47
One More Time…
60
30
2a
a
3a
45
45
a
aa2
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