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```Special Right Triangles
45-45-90
EQ: What are the relationships between
the sides on a 45 -45-90 triangle?
Moody Mathematics
Take a square…
Moody Mathematics
Find its diagonal
Here
it is
Moody Mathematics
Find its length
x
d
x
Moody Mathematics
x x d
2
2
2x  d
2
2x  d
2
2
2
2
x
d
x
x
2
2 d
x 2 d
Moody Mathematics
Summarize the
pattern:
Moody Mathematics
o
o
o
45 -45 -90
leg
leg
2
leg
Moody Mathematics
o
o
o
45 -45 -90
6
6
2
6
Moody Mathematics
o
o
o
45 -45 -90
8
8
2
8
Moody Mathematics
o
o
o
45 -45 -90
5
5
2
5
Moody Mathematics
o
o
o
45 -45 -90
10
10
2
10
Moody Mathematics
o
o
o
45 -45 -90
2
2
2
Moody Mathematics
o
o
o
45 -45 -90
3 2
6
3 2
Moody Mathematics
o
o
o
45 - 45 -90
4 2
4 2
8
Moody Mathematics
o
o
o
45 -45 -90
10 2
20
10 2
Moody Mathematics
The legs of 45 45 90 a triangle
are equal, the hypotenuse is
calculated immediately from the
equation c = a√2 . If the
hypotenuse value is given, the
side length will be equal to
a = c√2/2
Special Right
Triangles 30-60-90
Triangles
EQ: How do I find the lengths of sides of a
30-60-90 triangle
Now Let’s take
an Equilateral
Triangle…
Moody Mathematics
… Find its Altitude
Moody Mathematics
2
x
2
 x
2
2
  a  x 2
 2
x
2
2
a x 
4
2
2
4x x
2
x
xa  
4
4
a
2
3x
2
a 
4
x
2
x
Moody Mathematics
a
2
a  x2 3
4
a
2
x
a
x
2
x
3x

2
3x 4
4
x
2
x 3
a
2
Moody Mathematics
2
Summarize the
pattern:
Moody Mathematics
Shorter Leg
o
o
o
30 -60 -90
60
30
Longer leg
Moody Mathematics
½ Hyp.
o
o
o
30 -60 -90
60
30
½ Hyp.
3
Moody Mathematics
o
o
o
30 -60 -90
10
60
20
30
10 3
Moody Mathematics
In a 30-60-90 triangle, the ratio of
the sides is always in the ratio of
1:√3: 2. This is also known as the
30-60-90 triangle formula for
sides. y:y√3:2y.
Practice:
Moody Mathematics
o
o
o
30 -60 -90
14
60
28
30
14 3
Moody Mathematics
o
o
o
30 -60 -90
8
60
16
30
8 3
Moody Mathematics
o
o
o
30 -60 -90
9
60
18
30
9 3
Moody Mathematics
o
o
o
30 -60 -90
1
60
2
30
3
Moody Mathematics
o
o
o
30 -60 -90
3
60
6
30
3 3
Moody Mathematics
o
o
o
30 -60 -90
7 60
2
7
2
7
30
3
Moody Mathematics
o
o
o
30 -60 -90
2
60
4
30
2 3
Moody Mathematics
o
o
o
30 -60 -90
9 60
2
9
2
9
30
3
Moody Mathematics
o
o
o
30 -60 -90
60
4 3
2 3
30
6
Moody Mathematics
o
o
o
30 -60 -90
60
8 3
4 3
30
12
Moody Mathematics
Review Both
Patterns:
Moody Mathematics
o
o
o
45 -45 -90
leg
leg
2
leg
Moody Mathematics
½ Hyp.
o
o
o
30 -60 -90
60
30
½ Hyp.
3
Moody Mathematics
Mixed
Practice:
Moody Mathematics
12
60
24
30
12 3
Moody Mathematics
3
60
6
30
3 3
Moody Mathematics
8
8
2
8
Moody Mathematics
10
10
2
10
Moody Mathematics
1
60
2
30
3
Moody Mathematics
10
10
5 3
10
Moody Mathematics
15
60
30
30
15 3
Moody Mathematics
8
8
8 2
8
8
Moody Mathematics
8 3
30
16
8
60
Moody Mathematics
4 2
4 2
8
Moody Mathematics
3
60
6
30
3 3
Moody Mathematics
15
15
15 2
15
15
Moody Mathematics
60
10 3
5 3
30
15
Moody Mathematics
60
8 3
4 3
30
12
Moody Mathematics
4 2
4
4
Moody Mathematics
2
2
2
Moody Mathematics
18
18
9 3
18
Moody Mathematics
```
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