1) Special right triangles have properties that allow for determining side lengths without using the Pythagorean theorem. A 45-45-90 triangle has both legs that are congruent and the hypotenuse is √2 times the leg length. A 30-60-90 triangle has the hypotenuse that is 2 times the shorter leg and the longer leg is √3 times the shorter leg. 2) Examples are provided to demonstrate using the special right triangle properties to determine unknown side lengths. 3) Relationships between the legs and hypotenuse of 45-45-90 and 30-60-90 triangles are outlined to determine a leg from the hypotenuse or a longer leg from a shorter leg

1. 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
1 1 2
2
x
x
x
+ =
= + =
=

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
4 4 8
8
x
x
x
+ =
= + =
=
8 2 2.

4. 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 2
45°
45°

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

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

7. 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
2 2 2
 
 
= =  
 
  
x
16 2
8 2
2
= =
2.
16
2 8 2
2
= =
x

8. Examples
Find the value of x.
1.
2.
x
45° 20
x
5

9. Examples
Find the value of x.
1.
2.
x
45° 20 20
2 10 2
2
x = =
x
5 5
2
2
x =

10. 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.
Remember: the shorter leg is always opposite the smallest
(30°) angle; the longer leg is always opposite the 60°
angle.
3
x
2x
3
x
60°
30°

11. Examples
Find the value of x. Simplify radicals.
1. 2.
3. 4.
7
x
60°
30°
11
x
9
x
60°
16
16
60°
x

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

13. Examples
To find the shorter leg from the longer leg:
Find the value of x
1. 2.
9
x
60°
10
x
30°
longer leg 3 longer leg
3
3
3 3
 
 
=
 
 
  

14. Examples
To find the shorter leg from the longer leg:
Find the value of x
1. 2.
9
x
60°
10
x
30°
longer leg 3 longer leg
3
3
3 3
 
 
=
 
 
  
9
3 3 3
3
= =
x
10
3
3
x =