30-60-90 Triangles
Brenda Meery
Jen Kershaw
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both in
the U.S. and worldwide. Using an open-source, collaborative, and
web-based compilation model, CK-12 pioneers and promotes the
creation and distribution of high-quality, adaptive online textbooks
that can be mixed, modified and printed (i.e., the FlexBook®
textbooks).
Copyright © 2016 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/about/
terms-of-use.
Printed: April 5, 2016
AUTHORS
Brenda Meery
Jen Kershaw
www.ck12.org
C HAPTER
Chapter 1. 30-60-90 Triangles
1
30-60-90 Triangles
In this concept, you will learn about 30-60-90 right triangles. As Lindsay was walking near the Civic Center she
noticed a shadow on the ground. She looked at the shadow and then cast her eyes upward to see the top of a flagpole.
“That flagpole is quite tall but I bet I had to look up a longer distance than the pole is tall,” Lindsay said to herself.
“I would like to figure out the length of my line of sight from the ground to the top of the flagpole,” she added.
How can Lindsay use the diagram below to figure out the length of her line of sight?
In this concept, you will learn about 30◦ − 60◦ − 90◦ right triangles.
Equilateral Triangle
An equilateral triangle is one that has three sides of equal length and three angles of equal measure. The angles of
the triangle each measure 60◦ . When an altitude is drawn in an equilateral triangle, it bisects the angle from which
it is drawn and is a perpendicular bisector of the base. A perpendicular bisector of the base passes through the
midpoint of the base and is perpendicular to that side. The first diagram shows an equilateral triangle and the second
one shows an equilateral triangle and its altitude.
1
www.ck12.org
In the second triangle the line CM is the altitude of the equilateral triangle. The measure of 6 C has been divided
into two angles each measuring 30◦ . The altitude meets the base of the triangle AB at right angles and divides
the base into AM and MB which are equal in length. If AM + MB = AB then the length of AB = x + x = 2x. If
AB = AC = BC then AC = 2x and BC = 2x.
The altitude of the equilateral triangle has divided it into two equal right triangles. ∆BCM is a 30◦ −60◦ −90◦ triangle
and it will be used to show the relationship between the legs and the hypotenuse of this special right triangle.
Let’s begin by using the Pythagorean Theorem to find the measure of the altitude CM.
First, write down the values of (a, b, c) for the Pythagorean Theorem.
CM = a = ?
MB = b = x
BC = c = 2x
Next, write the Pythagorean Theorem and substitute in the values for (a, b, c).
c2 = a2 + b2
(2x)2 = a2 + (x)2
Next, do the indicated squaring on both sides of the equation.
(2x)2 = a2 + (x)2
(2x · 2x) = a2 + (x · x)
4x2 = a2 + x2
Next, isolate the variable by subtracting x2 from both sides of the equation and simplify.
4x2 = a2 + x2
= a2 + x2 − x2
3x2 = a2
4x2 − x2
Then, solve for the variable ’a’ by taking the square root of both sides of the equation.
3x2
√ √
3√
· x2
3√· x
x 3
The answer is x
2
= a√
=
a2
= a
= a
√
3.
√
If b = x then the answer can be expressed as b 3.
The relationship between the sides of the 30◦ − 60◦ − 90◦ triangle is:
• The length of the hypotenuse is twice the length of the shorter leg.
• The length of the shorter leg is √
one-half the length of the hypotenuse.
• The length of the longer leg is 3 times the length of the shorter leg.
Examples
Example 1
Earlier, you were given a problem about Lindsay and the flagpole shadow. She wants to know the length of her line
of sight to the top of the flagpole.
2
www.ck12.org
Chapter 1. 30-60-90 Triangles
Lindsay will have to use the relationships between the sides of a special triangle.
First, she should write down what she knows.
√
The length of the flagpole is the longer leg of the 30◦ − 60◦ − 90◦ triangle and is 7 3 feet in length.
The length of the shadow is the shorter leg and its length is unknown.
The length of the hypotenuse is the length of Lindsay’s line of sight and she wants to figure out this length.
Next, identify the variables.
Let ’b’ represent the length of the longer leg, ’a’ represent the length of the shorter leg and ’c’ represent her line of
sight.
Next, write down an equation showing the relationship between the two legs of the triangle.
b=
√
3·a
Next, fill any known values into the equation.
√
b
=
3·a
√
√
7 3 = a 3
Then, divide both sides of the equation by
√
3 to solve for the variable ’a’.
√
√
7 3 = a 3
√
√
7
√3 = a
√3
3
3
7 = a
The answer is 7.
The length of the shorter leg is 7 feet.
Next, write an equation showing the relationship between the hypotenuse and the shorter leg.
c = 2a
Next, substitute the length of the shorter leg into the equation.
c = 2a
c = 2(7)
Then, perform the multiplication on the right side of the equation.
c = 2(7)
c = 14
The answer is 14.
The length of Lindsay’s line of sight is 14 feet.
3
www.ck12.org
Example 2
For the following 30◦ − 60◦ − 90◦ triangle with a hypotenuse of 16 inches, find the length of the longer leg to the
nearest hundredth.
First, write down what you know from the diagram.
The hypotenuse is BC = 16 inches
The shorter leg is AC.
The longer leg is AB.
Next, write an equation that shows the relationship between the shorter leg and the hypotenuse.
1
AC = (BC)
2
Then, substitute the value for BC into the equation and simplify.
AC = 12 (BC)
AC = 12 (16)
AC = 8
The answer is 8.
The length of the shorter leg is 8 inches.
Now determine the length of the longer leg AB.
First, write an equation that shows the relationship between the two legs of the triangle.
AB = AC ·
√
3
Next, substitute the value of AC into the equation and simplify.
√
AB = AC · √3
AB = (8)
√· 3
AB = 8 3
4
www.ck12.org
Chapter 1. 30-60-90 Triangles
Then, use the calculator to express the length of the longer leg to the nearest hundredth.
√
AB = 8 3
AB = 8(1.73)
AB ≈ 13.8
The answer is approximately 13.8.
The length of the longer leg is approximately 13.8 inches.
Example 3
For the following 30◦ − 60◦ − 90◦ triangle with a longer leg of 6 inches, find the length of the shorter leg. Express
the length as a radical.
First, write down what you know from the diagram.
The longer leg is CD = 6 inches
The shorter leg is EC.
The hypotenuse is ED.
Next, write an equation that shows the relationship between the longer leg and the shorter leg.
CD =
√
3(EC)
Next, substitute the value for CD into the equation and simplify.
CD =
6 =
Next, divide both sides of the equation by
√
√3(EC)
3(EC)
√
3 to solve EC
√
3(EC)
√
3(EC)
= √
3
= EC
6 =
√6
3
√6
3
5
www.ck12.org
√
Next, multiply the left side of the equation by √3 to clear the radical from the denominator.
3
√6
= EC
3
√ √6 · √3
= EC
3
3
√
6√ 3
= EC
9
Next, simplify the left side of the equation.
√
6√ 3
= EC
√9
6 3
= EC
3
2
√
6 3 = EC
3√
2 3 = EC
√
The answer is 2 3.
√
The length of the shorter leg is 2 3 inches.
Example 4
For the following 30◦ − 60◦ − 90◦ triangle with a shorter leg of 4 feet, find the length of the hypotenuse to the nearest
tenth.
First, write down what you know from the diagram.
The hypotenuse is AB = ? feet
The shorter leg is AC = 4 feet
The longer leg is BC.
Next, write an equation that shows the relationship between the shorter leg and the hypotenuse.
2(AC) = AB
Then, substitute the value for AC into the equation and simplify.
6
www.ck12.org
Chapter 1. 30-60-90 Triangles
2(AC) = AB
2(4) = AB
8 = AB
The answer is 8.
The length of the hypotenuse is 8 feet.
Review
Find the missing length of the longer leg in each 30◦ − 60◦ − 90◦ triangle.
1. Short leg = 3
2. Short leg = 4
3. Short leg = 2
4. Short leg = 8
5. Short leg = 10
Use a calculator to figure out the approximate value of each longer leg. You may round your answer when necessary.
√
6. 3 3
√
7. 4 3
√
8. 2 3
√
9. 8 3
√
10. 10 3
Use what you have learned to solve each problem.
11. Janie had construction paper cut into and equilateral triangle. She wants to cut it into two smaller congruent
triangles. What will be the angle measurement of the triangles that result?
12. Madeleine has poster board in the shape of a square. She wants to cut two congruent triangles out of the poster
board without leaving any leftovers. What will be the angle measurements of the triangles that result?
√
13. A square window has a diagonal of 2 3 feet. What is the length of the shorter of its legs?
14. A square block of cheese is cut into two congruent wedges. If the shortest side of the original block was 9 inches,
how long is the diagonal cut?
15. Jerry wants to find the area of an equilateral triangle but only knows that the length of the shorter side is 4
centimeters. What is the height of Jerry’s triangle?
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 7.12.
Resources
7
www.ck12.org
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/169209
8