Geometry 9.4 Special Right Triangles. November 27, 2015Geometry 9.4 Special Right Triangles2 Goals I...
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![Page 1: Geometry 9.4 Special Right Triangles. November 27, 2015Geometry 9.4 Special Right Triangles2 Goals I will know the side lengths of special right triangles.](https://reader036.fdocuments.net/reader036/viewer/2022062803/56649f415503460f94c60a89/html5/thumbnails/1.jpg)
Geometry
9.4 Special Right Triangles
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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.
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 6
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April 21, 2023 Geometry 9.4 Special Right Triangles 7
Constructing Special Triangles
Construct an equilateral triangle.
Each angle measures ______.60
60 60
60
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 10
Constructing Special Triangles
This is called a 30 – 60 – 90 triangle.
60
30
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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
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 15
30-60-90 Triangle
60
30
2a
a
3a
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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
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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
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 21
30-60-90 Triangle
60
30
2a
a
3a
Learn the pattern!
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April 21, 2023 Geometry 9.4 Special Right Triangles 22
Try it. Find x and y.
60
30
x
15
y30
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April 21, 2023 Geometry 9.4 Special Right Triangles 23
Another one: Find h and y.
60
30
y
h14
7
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April 21, 2023 Geometry 9.4 Special Right Triangles 24
Again: Find x and y.
60
30
y
x
2
4
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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
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 29
45-45-90 Triangle
45
45
a
a
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April 21, 2023 Geometry 9.4 Special Right Triangles 30
Example 5
Solve for x & y.
45
45
5
xya
45
45a
a25
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 34
45-45-90 Triangle
45
45
a
a
Learn the pattern!
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April 21, 2023 Geometry 9.4 Special Right Triangles 35
Try it. Find x and y.
45
45
x
y8
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April 21, 2023 Geometry 9.4 Special Right Triangles 36
45-45-90 Triangle
12 2a
45
45
a
Shortcut pattern:
12 2a
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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
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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?
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April 21, 2023 Geometry 9.4 Special Right Triangles 39
Example 9 Solution
30 ft.
30 ft.
45
45
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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.
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April 21, 2023 Geometry 9.4 Special Right Triangles 41
Quick Quiz – 5 problems
30
xy
12
1. Find x & y.
24
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April 21, 2023 Geometry 9.4 Special Right Triangles 42
60
x
y
50
2. Find x & y.
25
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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
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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
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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
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April 21, 2023 Geometry 9.4 Special Right Triangles 46
5. Find the area.
45
7
7
17 7 24.5
2A
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April 21, 2023 Geometry 9.4 Special Right Triangles 47
One More Time…
60
30
2a
a
3a
45
45
a
aa2