Special Right Triangles

Draw 5 squares with each side length increasing by 11

2

3

4

5

Find the length of each diagonal1

2

3

4

5

Find the length of each diagonal1

a2 + b2 = c2 12 + 12 = c2

1 + 1 = c2

2 = c2

2 = c

Find the length of each diagonal1

2a2 + b2 = c2 22 + 22 = c2

4 + 4 = c2

8 = c2

8 = c 2 2 = c

Find the length of each diagonal1

2

3

a2 + b2 = c2 32 + 32 = c2

9 + 9 = c2

18 = c2

18 = c 3 2 = c

Find the length of each diagonal1

2

3

4

a2 + b2 = c2 42 + 42 = c2

16 + 16 = c2

32 = c2

32 = c 4 2 = c

Find the length of each diagonal1

2

3

4

5a2 + b2 = c2 52 + 52 = c2

25 + 25 = c2

50 = c2

50 = c 5 2 = c

Do you see the pattern1

2

3

4

5 5 2

4 2

3 2

2 2

2

Side length time square root of 2

Now, what do you know about squares and the triangle formed by

the diagonal.

• A square has 4 equal sides and 90° angles•The triangle is isosceles right

If the triangle is isosceles right, then what do you know about the base

angles?

45°

45°

An isosceles right triangle is a 45-45-90 triangle

45°

45°

45-45-90 triangles follow the same pattern we found in the squares.

45°

45°

a

a 2

So if you know one side you can find the other.

Find the missing sides

45°

45°

8

8 2c8

Find the missing sides

45°

45°

a

7 2

7

7

Find the missing sides

2525 2c

25

Find the missing sides

a

16 2

1616

45-45-90 triangles will have two basic diagrams that you will need to

recognize.

45°

45°

You will have to know 45-45-90 triangles… the hypotenuse = leg 2

45°

45°

OR.. the leg = hypotenuse 2

45°

45°

There is a second “special right triangle”…30-60-90

Obviously if one angle is 30° then the other is 60°

30°

Angle Sum of Triangle

30-60-90 triangle

The short leg (a) is always opposite the 30° angle

30°

a

30-60-90 triangleThe hypotenuse is always 2 times short leg

30°

2a

a

Draw five 30-60-90 trianglesEach short leg increasing by 1

30°

10

5

30°

8

4

30°6

3

30°

4

2

30°

2

1

Find the long legb

b

b

b b

Find the long leg

a2 + b2 = c2 12 + b2 = 22

1 + b2 = 4 b2 = 4-1 b2 = 3 b = 3 b = 3

30°

2

1

b

Find the long leg

a2 + b2 = c2 22 + b2 = 42

4 + b2 = 16 b2 = 16-4 b2 = 12 b = 12 b = 2 3

30°

4

2

30°

2

1

b

b

Find the long leg

a2 + b2 = c2 32 + b2 = 62

9 + b2 = 36 b2 = 36-9 b2 = 27 b = 27 b = 3 3

30°

4

2

30°

2

1

b

b30°6

3

b

Find the long leg

a2 + b2 = c2 42 + b2 = 82

16 + b2 = 64 b2 = 64-16 b2 = 48 b = 48 b = 4 3

30°

4

2

30°

2

1

b

b30°6

3

b 30°

8

4

b

Find the long leg30°

4

2

30°

2

1

b

b30°6

3

b 30°

8

4

b 30°

10

5

b

a2 + b2 = c2 52 + b2 = 102

25 + b2 = 100 b2 = 100-25 b = 75 b = 5 3

Do you see the pattern?

30°

2

30°

130°

3

3 3 30°

4

4 3

30°

5

5 3

2 3

3

Long leg = short leg time the square root of 3

30-60-90 triangle

30°

2a

a

a3

Find the missing side

30°

h

8

8316

b