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Obj. 24 Special Right Triangles
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
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