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Two Special Right Triangles
45°- 45°- 90°
30°- 60°- 90°
45°- 45°- 90°
The 45-45-90 triangle is based on the square with sides of 1 unit.
The 45-45-90 triangle is based on the square with sides of 1 unit.
1
1
1
1
45°- 45°- 90°
If we draw the diagonals we form two 45-45-90 triangles.
If we draw the diagonals we form two 45-45-90 triangles.
1
1
1
145°
45°
45°
45°
45°- 45°- 90°
Using the Pythagorean Theorem we can find the length of the diagonal.
Using the Pythagorean Theorem we can find the length of the diagonal.
1
1
1
145°
45°
45°
45°
45°- 45°- 90°
12 + 12 = c2
1 + 1 = c2
2 = c2
2 = c
12 + 12 = c2
1 + 1 = c2
2 = c2
2 = c
1
1
1
145°
45°
45°
45°
2
45°- 45°- 90°
Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1-2
Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1-2
1
1 2
45°
45°
45°- 45°- 90° Practice
4
4 2
SAME leg*2
445°
45°
45°- 45°- 90° Practice
9
9 2
SAME leg*2
945°
45°
45°- 45°- 90° Practice
2
2 2
SAME leg*2
245°
45°
45°- 45°- 90° Practice
14
SAME leg*2 7
745°
45°
45°- 45°- 90° Practice
45°- 45°- 90° Practice
3 2
hypotenuse2
45°
45°
45°- 45°- 90° Practice
3 2 2
= 3
45°- 45°- 90° Practice
3 2
hypotenuse2
45°
45°
3SAME
3
45°- 45°- 90° Practice
6 2
hypotenuse2
45°
45°
45°- 45°- 90° Practice
6 2 2
= 6
45°- 45°- 90° Practice
6 2
hypotenuse2
45°
45°
6SAME
6
45°- 45°- 90° Practice
11 2
hypotenuse2
45°
45°
45°- 45°- 90° Practice
11 2 2
= 11
45°- 45°- 90° Practice
112
hypotenuse2
45°
45°
11SAME
11
45°- 45°- 90° Practice
8
hypotenuse2
45°
45°
45°- 45°- 90° Practice
8 2
2 2 * = 82
2 = 42
45°- 45°- 90° Practice
8
hypotenuse2
45°
45°
42 SAME
42
45°- 45°- 90° Practice
4
hypotenuse2
45°
45°
45°- 45°- 90° Practice
4 2
2 2 * = 42
2 = 22
45°- 45°- 90° Practice
4
hypotenuse2
45°
45°
22 SAME
22
45°- 45°- 90° Practice
6
Hypotenuse 2
45°
45°
45°- 45°- 90° Practice
6 2
2 2 * = 62
2 = 32
45°- 45°- 90° Practice
6
hypotenuse2
45°
45°
32 SAME
32
30°- 60°- 90°
The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.
The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.
22
2
60° 60°
22
2
60° 60°
30°- 60°- 90°
The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles.
The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles.
11
30°30°
30°
60°
This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg.
This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg.
30°- 60°- 90°
hypotenuse
hypotenuseShort LegShort Leg
Long
Leg
Long
Leg
60°
30°
30°- 60°- 90° Practice
1
2
We saw that the hypotenuse is twice the short leg.
We saw that the hypotenuse is twice the short leg. We can use the Pythagorean Theorem to find the long leg.
We can use the Pythagorean Theorem to find the long leg.
60°
30°
30°- 60°- 90° Practice
1
2 3
A2 + B2 = C2
A2 + 12 = 22
A2 + 1 = 4
A2 = 3
A = 3
A2 + B2 = C2
A2 + 12 = 22
A2 + 1 = 4
A2 = 3
A = 3
30°- 60°- 90°
Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 3
Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 3
60°
30°
3
1
2
60°
30°
30°- 60°- 90° Practice
4
8
Hypotenuse = short leg * 2
43
The key is to find the length of the short side.
The key is to find the length of the short side.
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
5
10 Hypotenuse = short leg * 2 53
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
7
14 Hypotenuse = short leg * 2 73
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
3
23Hypotenuse = short leg * 2 3
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
10
210Hypotenuse = short leg * 2
30
Long Leg = short leg * 3
30°- 60°- 90° Practice
60°
30°
30°- 60°- 90° Practice
11
22 Short Leg = Hypotenuse 2
113
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
2
4 Short Leg = Hypotenuse 2
23
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
9
18 Short Leg = Hypotenuse 2
93
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
15
30 Short Leg = Hypotenuse 2
153
Long Leg = short leg * 3
60°
30°
30°- 60°- 90° Practice
23
46 Hypotenuse = Short Leg * 2 233
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
14
28 Hypotenuse = Short Leg * 2 143
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
16
32 Hypotenuse = Short Leg * 2 163
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
3 3
63Hypotenuse = Short Leg * 2 9
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
4 3
83Hypotenuse = Short Leg * 2 12
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
9 3
183Hypotenuse = Short Leg * 2 27
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
7 3
143Hypotenuse = Short Leg * 2 21
Short Leg = Long leg 3
60°
30°
30°- 60°- 90° Practice
113
223Hypotenuse = Short Leg * 2 33
Short Leg = Long leg 3
