30-60-90 Right Triangles
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: July 10, 2016
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. 30-60-90 Right Triangles
1
30-60-90 Right Triangles
Here you’ll learn that the sides of a 30-60-90 right triangle are in the ratio x : x
√
3 : 2x.
30-60-90 Right Triangles
One of the two special right triangles is called a 30-60-90 triangle, after its three angles.
√
30-60-90 Theorem: If a triangle has angle measures 30◦ , 60◦ and 90◦ , then the sides are in the ratio x : x 3 : 2x.
√
The shorter leg is always x, the longer leg is always x 3, and the hypotenuse is always 2x. If you ever forget these
theorems, you can still use the Pythagorean Theorem.
What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the
lengths of its other sides?
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136654
Examples
Example 1
Find the value of x and y.
We are given the longer leg.
x
√
3 = 12
√
√
√
12
3 12 3
x= √ · √ =
=4 3
3
3
3
The hypotenuse is
√
√
y = 2(4 3) = 8 3
1
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Example 2
Find the value of x and y.
We are given the hypotenuse.
2x = 16
x=8
The longer leg is
√
√
y = 8· 3 = 8 3
Example 3
Find the length of the missing sides.
√
We are given the shorter leg. If x = 5, then the longer leg, b = 5 3, and the hypotenuse, c = 2(5) = 10.
Example 4
Find the length of the missing sides.
We are given the hypotenuse. 2x = 20, so the shorter leg, f =
2
20
2
√
= 10, and the longer leg, g = 10 3.
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Chapter 1. 30-60-90 Right Triangles
Example 5
√
A rectangle has sides 4 and 4 3. What is the length of the diagonal?
If you are not given a picture, draw one.
The two lengths are x, x
√
3, so the diagonal would be 2x, or 2(4) = 8.
If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.
√ 2
42 + 4 3 = d 2
16 + 48 = d 2
√
d = 64 = 8
Review
1. In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
2. In a 30-60-90 triangle, if the shorter leg is x, then the longer leg is __________ and the hypotenuse is ___________.
√
3. A rectangle has sides of length 6 and 6 3. What is the length of the diagonal?
4. Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides?
For questions 5-12, find the lengths of the missing sides. Simplify all radicals.
5.
6.
7.
3
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8.
9.
10.
11.
12.
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.6.
Resources
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1362
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1363
4
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Chapter 1. 30-60-90 Right Triangles
5