Two Special Right Triangles

45°- 45°- 90°30°- 60°- 90°

HW: Special Right Triangles WS1(side 1 only: 45-45-90)

45°- 45°- 90°

The 45-45-90 triangle is based on the square with sides of 1 unit.

11

11

11

11

45°- 45°- 90°

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.

1

1

1

145°

45°

45°

45°

45°- 45°- 90°

1

1

1

145°

45°

45°

45°

2

45°- 45°- 90°

1

1

45°

45°

In a 45° – 45° – 90° triangle the hypotenuse is the

square root of two times as long as each leg

Rule:

45°- 45°- 90° Practice

4SAME

445°

45°

45°- 45°- 90° Practice

9SAME

945°

45°

45°- 45°- 90° Practice

SAME45°

45°

45°- 45°- 90° Practice

45°- 45°- 90° Practice

45°

45°

45°- 45°- 90° Practice

= 3

45°- 45°- 90° Practice

45°

45°

3SAME

3

45°- 45°- 90° Practice

45°

45°

45°- 45°- 90° Practice

= 11

45°- 45°- 90° Practice

45°

45°

11SAME

11

45°- 45°- 90° Practice

8

45°

45°

45°- 45°- 90° Practice

8 * =

2

Rationalize the denominator

45°- 45°- 90° Practice

8

45°

45°

SAME

45°- 45°- 90° Practice

4

45°

45°

45°- 45°- 90° Practice

4 * =

2

Rationalize the denominator

45°- 45°- 90° Practice

4

45°

45°

SAME

45°- 45°- 90° Practice

7

45°

45°

45°- 45°- 90° Practice

7 *

Rationalize the denominator

45°- 45°- 90° Practice

7

45°

45°

SAME

Find the value of each variable. Write answers in simplest radical form.

• Know the basic triangles

• Set known information equal to the corresponding part of the basic triangle

• Solve for the other sides

10 2 x x 5 2 y 5 2

Find the value of each variable.Write the answers in simplest radical form.

Find the value of each variable. Write answers in simplest radical form.

Two Special Right Triangles

45°- 45°- 90°

30°- 60°- 90°

HW: Special Right Triangles WS1(side 2 only: 30-60-90)

30°- 60°- 90°The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.

2222

22

60° 60°

22

2

60° 60°

30°- 60°- 90°

The altitude 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.

30°- 60°- 90°

hypotenuseShort Leg

Long

Leg

60°

30°

30°- 60°- 90° Practice

1

2

We saw that the hypotenuse is twice

the short leg.

We can use the Pythagorean

Theorem to find the long leg.

60°

30°

30°- 60°- 90° Practice

1

2

30°- 60°- 90°

60°

30°

1

2

30° – 60° – 90° TriangleIn a 30° – 60° –

90° triangle, the hypotenuse is twice as

long as the shorter leg, and the longer leg is the square root of

three times as long as the shorter leg

30°-60°-90°

60°

30°

30°- 60°- 90° Practice

4

8

Hypotenuse = short leg * 2

The key is to find the length of the

short side.

60°

30°

30°- 60°- 90° Practice

5

10

hyp = short leg * 2

60°

30°

30°- 60°- 90° Practice

7

14

* 2

60°

30°

30°- 60°- 90° Practice

3

* 2

60°

30°

30°- 60°- 90° Practice

* 2

30°- 60°- 90° Practice

60°

30°

30°- 60°- 90° Practice

11

22

Short Leg = hyp 2

60°

30°

30°- 60°- 90° Practice

2

4

60°

30°

30°- 60°- 90° Practice

9

18

60°

30°

30°- 60°- 90° Practice

23

46

60°

30°

30°- 60°- 90° Practice

14

28

60°

30°

30°- 60°- 90° Practice

9

60°

30°

30°- 60°- 90° Practice

hyp = Short Leg * 2

12

60°

30°

30°- 60°- 90° Practice

27

hyp = Short Leg * 2

60°

30°

30°- 60°- 90° Practice

20

hyp = Short Leg * 2

60°

30°

30°- 60°- 90° Practice

33

hyp = Short Leg * 2

PRACTICE

Find all the missing sides for each triangle.

Solving Strategy• Know the basic triangles• Set known information equal to the

corresponding part of the basic triangle• Solve for the other sides

Find the value of each variable. Write answers in simplest radical form.

Find the value of each variable. Write answers in simplest radical form.

30

60

12 3

2412

30

60

5.5.55

5.55.5

5.5.55

2

1010 1010

1010

2

30

60

3

21

5

1010

10

30

60

8 8 2

30

60

30

60

3 3

63

15

15

15 2

15

15

30

60

12

8 34 3

88

2 22

2

18

18

189 3

Find the distance across the canyon.

30-60-90 b

Find the length of the canyon wall (from the edge to the river).

b

cc = b * 2

Is it more or less than a mile across the canyon?

5280 ft = 1 mile