special right triangles 45-45-90 & 30-60-90. special righttriangles there are two types of...
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Special Right Triangles45-45-90 & 30-60-90
Special RightTriangles
There are two types of Special Right Triangles. Each one has a standard set of rules.
These triangles show up on the SAT and every standardized test.
45
45 60
30
Special Right Triangles
Find the missing sides of each triangle.
45
45
45
4545
6 zx
w
y
mm
43 2
6 2
=6
2 2 2
2
2
4 4
16 16
32
32
16 2
4 2
z
z
z
z
z
z
4 2
2 2 2
2
2
6 6
36 36
72
72
36 2
6 2
x
x
x
x
x
x
2 2 2
2
2
2
2
2
(3 2)
2 3 2 3 2
2 9 4
2 9 2
2 18
9
3
m m
m
m
m
m
m
m
=4=3=3
Each 45-45-90 triangle is Isosceles
45
90
Fill in the missing angle
Special Right Triangles
So, based on our examples, let’s come up with some shortcuts for the 45-45-90 triangle…
example:45
45
6
6
6 245
45
hyp
leg
leg
SAME2
2
2
2
Special Note:You decide to multiply or divide based on whether the side you are going to is larger or smaller
So how do you get from the leg to the hypotenuse?
6 ____ 6 22
Special Right Triangles
Find x and y. Leave answers in SRF
45
45
10=16
x
x
y
8 2
y
=10
10 28 2
Same
2
Same 2
(8 2) 2
8 4
8 2
16
Start by labeling sides.
Special Right Triangles
Find x and y. Leave answers in SRF
45
45
x x
y
10 62
Same
10 3
10 3
(10 6) 2
10 6
2
10 3
6
y
23 2
6 6 2 6 2 6 23 2
22 2 2 4
Same
3 2
Start by labeling sides.
Special Right Triangles
The 30-60-90 triangle has its own rules…
Let’s see if we can find x and y.
60
30
5
xy
5cos60
5.5
.5 5
10
y
y
y
y
Find y using trigonometry and the 60 degree angle
Find x using the Pythagorean Theorem
2 2 2
2
2
5 10
25 100
75
75
25 3
5 3
x
x
x
x
x
x
=105 3
Special Right Triangles
So, based on our example, let’s come up with some shortcuts for 30-60-90 triangles
60
30
5
105 330
60
hypLong leg
Short leg
2
2
3
3
Special Note:You choose to multiply or divide based on whether the side you are going to is larger or smaller
Special Right Triangles
Fin x and y. Leave answers in SRF
60
30
60
30
x
y
x15
y
20
Start by labeling sides.
Long Leg
Short Leg
Hypotenuse2
3
10 3
Remember that you decide to multiply or divide based on whether you are going to a larger or smaller side
3
15 15 3 15 3 15 35 3
33 3 3 9
=10 5 3
2
10 3Long Leg
Short Leg
Hypotenuse
Special Right Triangles
Find the Area of the figure.
Area = (base)(height)
= (10)(height)
= (10)
square units60
66
10
10
To find the height, we need to draw in a triangle and use our special triangle rules
h
60
6 h
2
3
3
3 3
(3 3)
30 3
3 3
Special Right Triangles
Find the Area of the Equilateral Triangle.
Area = ½ (base)(height)
= ½ (8)(height)
= ½ (8)
square units
To find the height, we need to draw in a triangle and use our special triangle rules
3
16 3
8
8
8 h
8
4
h
You can use the Pythagorean Theorem, but this is a 30-60-90 triangle because the triangle above is equilateral..
30
60
4 3
4 3