Obj. 24 Special Right Triangles

The student is able to (I can):

Identify when a triangle is a 45-45-90 or 30-60-90 triangle

Use special right triangle relationships to solve problems

Consider the following triangle:

To find x, we would use a2 + b2 = c2, which gives us:

What would x be if each leg was 2?

1

1 x

2 2 2

2

1 1 x

x 1 1 2

x 2

+ =

= + =

=

Again, we will use the Pythagorean Theorem

Simplifying the radical, we can factor to give us

Do you notice a pattern?

2

2 x

2 2 2

2

2 2 x

x 4 4 8

x 8

+ =

= + =

=

82 2.

Thm 5-8-1 45-45-90 Triangle Theorem

In a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is times the length of the leg.

2

x

x x 245

45

Example Find the value of x. Give your answer in simplest radical form.

1.

2.

3.

45

x

8

8 2

x7

7 2

9 2x

9 29

2=

If we know the hypotenuse and need to find the leg of a 45-45-90 triangle, we have to divide by . This means we will have to rationalize the denominator, which means to multiply the top and bottom by the radical.

The shortcut for this is to divide the hypotenuse by 2 and then multiply by

2

16 x

16 16 2x

2 2 2

= =

16 28 2

2= =

2.

16x 2 8 2

2= =

Examples Find the value of x.

1.

2.

x

45 20

20x 2 10 2

2= =

x

5

5x 2

2=

Thm 5-8-2 30-60-90 Triangle Theorem

In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg.

Note: the shorter leg is always opposite the 30 angle; the longer leg is always opposite the 60 angle.

3

x

2xx 3

60

30

Examples Find the value of x. Simplify radicals.

1. 2.

3. 4.

7

x

60

30

11x

9

x

60

1616

60

x

9 316

3 8 32

=

1411

5.52

=

Examples

To find the shorter leg from the longer leg:

Find the value of x

1.

2.

9

x60

10

x

30

longer leg 3 longer leg3

3 3 3

=

9x 3 3 3

3= =

10x 3

3=