Identify when a triangle is a 45-45-90 or 30-60-90 triangle Use special right triangle relationships to solve problems

1. 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

2. Consider the following triangle:
x
1
1
To find x, we would use a2 + b2 = c2, which
gives us:
12 + 12 = x 2
x2 = 1 + 1 = 2
x= 2
What would x be if each leg was 2?

3. x
2
2
Again, we will use the Pythagorean Theorem
22 + 22 = x 2
x2 = 4 + 4 = 8
x= 8
Simplifying the radical, we can factor 8
to give us 2 2.
Do you notice a pattern?

4. Thm 5-8-1
45º-45º-90º Triangle Theorem
In a 45º-45º-90º triangle, both legs are
congruent, and the length of the
hypotenuse is 2 times the length of
the leg.
x 45º
x 2
45º
x

5. Example
Find the value of x. Give your answer in
simplest radical form.
x
1. 8 2
45º
8
2. 7 2
9 2
=9
3.
2
x
x
7
9 2

6. If we know the hypotenuse and need to find
the leg of a 45-45-90 triangle, we have to
divide by 2 . This means we will have to
rationalize the denominator, which means
to multiply the top and bottom by the
radical.
16  16   2 
x=
=


2  2  2 
16 2
=
=8 2
2
16
x
The shortcut for this is to divide the
hypotenuse by 2 and then multiply by 2.
16
x=
2 =8 2
2

7. Examples
Find the value of x.
1. x =
20
2 = 10 2
2
45º
20
x
5
2. x =
2
2
x
5

8. Thm 5-8-2
30º-60º-90º Triangle Theorem
In a 30º-60º-90º triangle, the length of
the hypotenuse is 2 times the length of
the shorter leg, and the length of the
longer leg is 3 times the length of the
shorter leg.
30º
x 3
2x
60º
x
Note: the shorter leg is always opposite
the 30º angle; the longer leg is always
opposite the 60º angle.

9. Examples
Find the value of x. Simplify radicals.
1. 14
2.
x
11
= 5.5
2
11
x
30º
60º
7
3. 9 3
4.
16
3 =8 3
2
16
16
x
x
60º
9
60º

10. To find the shorter leg from the longer leg:
 longer leg   3  longer leg
3


=
3  3 
3

Examples
Find the value of x
1. x =
9
3 =3 3
3
9
60º
x
10
3
2. x =
3
10
30º
x