Transcript
Page 1: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

Two Special Right Triangles

45°- 45°- 90°

30°- 60°- 90°

Page 2: 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

Page 3: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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°

Page 4: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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°

Page 5: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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

Page 6: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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°

Page 7: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

4

4 2

SAME leg*2

445°

45°

Page 8: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

9

9 2

SAME leg*2

945°

45°

Page 9: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

2

2 2

SAME leg*2

245°

45°

Page 10: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

14

SAME leg*2 7

745°

45°

Page 11: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

Page 12: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

3 2

hypotenuse2

45°

45°

Page 13: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

3 2 2

= 3

Page 14: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

3 2

hypotenuse2

45°

45°

3SAME

3

Page 15: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6 2

hypotenuse2

45°

45°

Page 16: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6 2 2

= 6

Page 17: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6 2

hypotenuse2

45°

45°

6SAME

6

Page 18: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

11 2

hypotenuse2

45°

45°

Page 19: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

11 2 2

= 11

Page 20: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

112

hypotenuse2

45°

45°

11SAME

11

Page 21: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

8

hypotenuse2

45°

45°

Page 22: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

8 2

2 2 * = 82

2 = 42

Page 23: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

8

hypotenuse2

45°

45°

42 SAME

42

Page 24: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

4

hypotenuse2

45°

45°

Page 25: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

4 2

2 2 * = 42

2 = 22

Page 26: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

4

hypotenuse2

45°

45°

22 SAME

22

Page 27: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6

Hypotenuse 2

45°

45°

Page 28: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6 2

2 2 * = 62

2 = 32

Page 29: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

45°- 45°- 90° Practice

6

hypotenuse2

45°

45°

32 SAME

32

Page 30: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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°

Page 31: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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°

Page 32: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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

Page 33: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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.

Page 34: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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

Page 35: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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

Page 36: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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

Page 37: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

5

10 Hypotenuse = short leg * 2 53

Long Leg = short leg * 3

Page 38: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

7

14 Hypotenuse = short leg * 2 73

Long Leg = short leg * 3

Page 39: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

3

23Hypotenuse = short leg * 2 3

Long Leg = short leg * 3

Page 40: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

10

210Hypotenuse = short leg * 2

30

Long Leg = short leg * 3

Page 41: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

30°- 60°- 90° Practice

Page 42: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

11

22 Short Leg = Hypotenuse 2

113

Long Leg = short leg * 3

Page 43: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

2

4 Short Leg = Hypotenuse 2

23

Long Leg = short leg * 3

Page 44: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

9

18 Short Leg = Hypotenuse 2

93

Long Leg = short leg * 3

Page 45: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

15

30 Short Leg = Hypotenuse 2

153

Long Leg = short leg * 3

Page 46: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

23

46 Hypotenuse = Short Leg * 2 233

Short Leg = Long leg 3

Page 47: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

14

28 Hypotenuse = Short Leg * 2 143

Short Leg = Long leg 3

Page 48: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

16

32 Hypotenuse = Short Leg * 2 163

Short Leg = Long leg 3

Page 49: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

3 3

63Hypotenuse = Short Leg * 2 9

Short Leg = Long leg 3

Page 50: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

4 3

83Hypotenuse = Short Leg * 2 12

Short Leg = Long leg 3

Page 51: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

9 3

183Hypotenuse = Short Leg * 2 27

Short Leg = Long leg 3

Page 52: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

7 3

143Hypotenuse = Short Leg * 2 21

Short Leg = Long leg 3

Page 53: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

60°

30°

30°- 60°- 90° Practice

113

223Hypotenuse = Short Leg * 2 33

Short Leg = Long leg 3

Page 54: Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

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