Geometry (B): READ: Converse of Pythagorean Theorem Reading

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:

$a^2 + b^2 = c^2$

The Converse of the Pythagorean Theorem is also true. That is, if the lengths of three sides of a triangle make the equation $a^2+b^2=c^2$ 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.

$a^2 + b^2 &= c^2\\ 8^2 + 15^2 &= 17^2\\ 64 + 225 &= 289\\ 289 &= 289$

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 $c$ , $a^2 + b^2 = c^2$ . The converse of the Pythagorean Theorem states that if $a^2 + b^2 = c^2$ , 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 $90^\circ$ ). In symbols, if $a^2+b^2>c^2$ 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 $a^2+b^2 > c^2$ 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.

$8^2 + 13^2 &= c^2\\ 64+169 &=c^2\\ 233&=c^2$

The sum of the squares of the two shorter legs is $233.$ Compare this to the square of the longest side, $15.$

$15^2=225$

The square of the longest side is $225.$ Since $8^2+13^2=233\neq 255=15^2$ , 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 $90^\circ$ ) 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 $a^2+b^2<c^2$ , 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.

$a^2 + b^2 &= 5^2+6^2\\ &= 25+36\\ &= 61$

The sum of the squares of the two shorter legs is $61.$ Compare this to the square of the longest side, $10.$

$10^2=100$

The square of the longest side is $100$ . Since $5^2+6^2 \neq 100^2$ , 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 $a \le b < c$ . Be sure to use the longest side for the hypotenuse.

If $a^2 + b^2 = c^2$ , the figure is a right triangle.
If $a^2 + b^2 > c^2$ , the figure is an acute triangle.
If $a^2 + b^2 < c^2$ , 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.

$a^2+b^2 &= 9^2 + 11^2\\ &= 81 + 121 \\ &= 202$

The sum of the squares of the two shorter legs is $202.$ Compare this to the square of the longest side, $14.$

$14^2=196$

The square of the longest side is $196.$ Therefore, the two values are not equal, $a^2+b^2\neq c^2$ and this triangle is not a right triangle. Compare the two values, $a^2+b^2$ and $c^2$ to identify which is greater.

$202 > 196$

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.

$a^2+b^2 &= 16^2 + 30^2\\ &= 256+900\\ &=1156$

The sum of the squares of the two legs is $1,156$ . Compare this to the square of the longest side, $34$ .

$c^2=34^2=1156$

The square of the longest side is $1,156.$ Since these two values are equal, $a^2+b^2=c^2$ , 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.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

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.

$a = 9\;\mathrm{in},\ b = 12\;\mathrm{in},\ c = 15\;\mathrm{in}$
$a = 7\;\mathrm{cm},\ b = 7\;\mathrm{cm}, c = 8\;\mathrm{cm}$
$a = 4\;\mathrm{m},\ b = 8\;\mathrm{m},\ c = 10\;\mathrm{m}$
$a = 10\;\mathrm{ft},\ b = 22\;\mathrm{ft}, c = 23\;\mathrm{ft}$
$a = 21\;\mathrm{cm},\ b = 28\;\mathrm{cm},\ c = 35\;\mathrm{cm}$
$a = 10\;\mathrm{ft},\ b = 12\;\mathrm{ft},\ c = 14\;\mathrm{ft}$
$a = 15\;\mathrm{m},\ b = 18\;\mathrm{m},\ c = 30\;\mathrm{m}$
$a = 5\;\mathrm{in},\ b = \sqrt{75}\;\mathrm{in},\ c = 110\;\mathrm{in}$
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?
1. $m\angle A=$
2. $m\angle B=$

Review Answers

Right
Acute
Obtuse
Acute
Right
Acute
Obtuse
Obtuse
The side with length $\sqrt 13$ should be the hypotenuse since it is the longest side. The order of the legs does not matter
$m\angle A=45^\circ , m\angle B=90^\circ$

Last modified: Tuesday, June 29, 2010, 10:22 AM