Converse of the Pythagorean
Theorem
CK12 Editor
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-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2013 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/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License
(http://creativecommons.org/licenses/by-nc-sa/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/terms.
Printed: July 7, 2013
AUTHOR
CK12 Editor
www.ck12.org
C ONCEPT
Concept 1. Converse of the Pythagorean Theorem
1
Converse of the
Pythagorean Theorem
Learning Objectives
•
•
•
•
Understand the converse of the Pythagorean Theorem.
Identify acute triangles from side measures.
Identify obtuse triangles from side measures.
Classify triangles in a number of different ways.
Converse of the Pythagorean Theorem
In the last lesson, you learned about the Pythagorean Theorem and how it can be used. As you recall, it states that
the sum of the squares of the legs of any right triangle will equal the square of the hypotenuse. If the lengths of the
legs are labeled a and b, and the hypotenuse is c, then we get the familiar equation:
a2 + b2 = c2
The Converse of the Pythagorean Theorem is also true. That is, if the lengths of three sides of a triangle make the
equation a2 + b2 = c2 true, then they represent the sides of a right triangle.
With this converse, you can use the Pythagorean Theorem to prove that a triangle is a right triangle, even if you do
not know any of the triangle’s angle measurements.
Example 1
Does the triangle below contain a right angle?
This triangle does not have any right angle marks or measured angles, so you cannot assume you know whether the
triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they
are related. Two of the sides (15 and 17) are relatively close in length. The third side (8) is about half the length of
the two longer sides.
To see if the triangle might be right, try substituting the side lengths into the Pythagorean Theorem to see if they
makes the equation true. The hypotenuse is always the longest side, so 17 should be substituted for c. The other two
values can represent a and b and the order is not important.
1
www.ck12.org
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289
Since both sides of the equation are equal, these values satisfy the Pythagorean Theorem. Therefore, the triangle
described in the problem is a right triangle.
In summary, example 1 shows how you can use the converse of the Pythagorean Theorem. The Pythagorean Theorem
states that in a right triangle with legs a and b, and hypotenuse c, a2 + b2 = c2 . The converse of the Pythagorean
Theorem states that if a2 + b2 = c2 , then the triangle is a right triangle.
Identifying Acute Triangles
Using the converse of the Pythagorean Theorem, you can identify whether triangles contain a right angle or not.
However, if a triangle does not contain a right angle, you can still learn more about the triangle itself by using the
formula from Pythagorean Theorem. If the sum of the squares of the two shorter sides of a triangle is greater than
the square of the longest side, the triangle is acute (all angles are less than 90◦ ). In symbols, if a2 + b2 > c2 then the
triangle is acute.
Identifying the "shorter" and "longest" sides may seem ambiguous if sides have the same length, but in this case
any ordering of equal length sides leads to the same result. For example, an equilateral triangle always satisfies
a2 + b2 > c2 and so is acute.
Example 2
Is the triangle below acute or right?
The two shorter sides of the triangle are 8 and 13. The longest side of the triangle is 15. First find the sum of the
squares of the two shorter legs.
82 + 132 = c2
64 + 169 = c2
233 = c2
The sum of the squares of the two shorter legs is 233. Compare this to the square of the longest side, 15.
2
www.ck12.org
Concept 1. Converse of the Pythagorean Theorem
152 = 225
The square of the longest side is 225. Since 82 + 132 = 233 6= 255 = 152 , this triangle is not a right triangle. Compare
the two values to identify which is greater.
233 > 225
The sum of the square of the shorter sides is greater than the square of the longest side. Therefore, this is an acute
triangle.
Identifying Obtuse Triangles
As you have probably figured out, you can prove a triangle is obtuse (has one angle larger than 90◦ ) by using a
similar method. Find the sum of the squares of the two shorter sides in a triangle. If this value is less than the square
of the longest side, the triangle is obtuse. In symbols, if a2 + b2 < c2 , then the triangle is obtuse. You can solve this
problem in a manner almost identical to example 2 above.
Example 3
Is the triangle below acute or obtuse?
The two shorter sides of the triangle are 5 and 6. The longest side of the triangle is 10. First find the sum of the
squares of the two shorter legs.
a2 + b2 = 52 + 62
= 25 + 36
= 61
The sum of the squares of the two shorter legs is 61. Compare this to the square of the longest side, 10.
102 = 100
3
www.ck12.org
The square of the longest side is 100. Since 52 + 62 6= 1002 , this triangle is not a right triangle. Compare the two
values to identify which is greater.
61 < 100
Since the sum of the square of the shorter sides is less than the square of the longest side, this is an obtuse triangle.
Triangle Classification
Now that you know the ideas presented in this lesson, you can classify any triangle as right, acute, or obtuse given
the length of the three sides. Begin by ordering the sides of the triangle from smallest to largest, and substitute the
three side lengths into the equation given by the Pythagorean Theorem using a ≤ b < c. Be sure to use the longest
side for the hypotenuse.
• If a2 + b2 = c2 , the figure is a right triangle.
• If a2 + b2 > c2 , the figure is an acute triangle.
• If a2 + b2 < c2 , the figure is an obtuse triangle.
Example 4
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are 9 and 11. The longest side of the triangle is 14. First find the sum of the
squares of the two shorter legs.
a2 + b2 = 92 + 112
= 81 + 121
= 202
The sum of the squares of the two shorter legs is 202. Compare this to the square of the longest side, 14.
142 = 196
The square of the longest side is 196. Therefore, the two values are not equal, a2 + b2 6= c2 and this triangle is not a
right triangle. Compare the two values, a2 + b2 and c2 to identify which is greater.
4
www.ck12.org
Concept 1. Converse of the Pythagorean Theorem
202 > 196
Since the sum of the square of the shorter sides is greater than the square of the longest side, this is an acute triangle.
Example 5
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are 16 and 30. The longest side of the triangle is 34. First find the sum of the
squares of the two shorter legs.
a2 + b2 = 162 + 302
= 256 + 900
= 1156
The sum of the squares of the two legs is 1, 156. Compare this to the square of the longest side, 34.
c2 = 342 = 1156
The square of the longest side is 1, 156. Since these two values are equal, a2 + b2 = c2 , and this is a right triangle.
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations.
Specifically, we have learned:
•
•
•
•
How to use the converse of the Pythagorean Theorem to prove a triangle is right.
How to identify acute triangles from side measures.
How to identify obtuse triangles from side measures.
How to classify triangles in a number of different ways.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting
ways to apply the Pythagorean Theorem and its converse to mathematical situations.
5
www.ck12.org
Points to Consider
Use the Pythagorean Theorem to explore relationships in common right triangles. Do you find that the sides are
proportionate?
Review Questions
Solve each problem.
For exercises 1-8, classify the following triangle as acute, obtuse, or right based on the given side lengths. Note, the
figure is not to scale.
1.
2.
3.
4.
5.
6.
7.
8.
9.
a = 9 in, b = 12 in, c = 15 in
a = 7 cm, b = 7 cm, c = 8 cm
a = 4 m, b = 8 m, c = 10 m
a = 10 ft, b = 22 ft, c = 23 ft
a = 21 cm, b = 28 cm, c = 35 cm
a = 10 ft, b = 12 ft, c = 14 ft
a = 15 m, b =√18 m, c = 30 m
a = 5 in, b = 75 in, c = 110 in
In the triangle below, which sides should you use for the legs (usually called sides a , and b) and the hypotenuse
(usually called side c), in the Pythagorean theorem? How do you know?
10.
6
www.ck12.org
Concept 1. Converse of the Pythagorean Theorem
a. m6 A =
b. m6 B =
Review Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
Right
Acute
Obtuse
Acute
Right
Acute
Obtuse
Obtuse
√
The side with length 13 should be the hypotenuse since it is the longest side. The order of the legs does not
matter
10. m6 A = 45◦ , m6 B = 90◦
7