Identify special right triangles and use their properties to solve problems.

1. 4.11.2 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:
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
+ =
= + =
=

3. 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
+ =
= + =
=
8
2 2.

4. 45º-45º-90º Triangle Theorem
In a 45º-45º-90º triangle, both legs are
congruent, and the length of the
hypotenuse is times the length of
the leg.
2
x
x x 245º
45º

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

6. 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 2
x
2 2 2
  
= =   
  
16 2
8 2
2
= =
2.
16
x 2 8 2
2
= =

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

8. 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 times the length of the
shorter leg.
Remember: the shorter leg is always
opposite the smallest (30º) angle; the
longer leg is always opposite the 60º
angle.
3
x
2xx 3
60º
30º

9. Examples Find the value of x. Simplify radicals.
1. 2.
3. 4.
7
x
60º
30º
11x
9
x
60º
1616
60º
x
9 3
16
3 8 3
2
=
14
11
5.5
2
=

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