11.8: Pythagorean Theorem and its Converse
What if you were told that the side lengths of a triangle were 4, 5, and 7? How could you determine if the triangle were a right triangle? After completing this Concept, you'll be able to use the Pythagorean Theorem and its converse to solve problems like this one.
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CK12 Foundation: The Pythagorean Theorem and its Converse
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The Pythagorean Theorem can also be used to find the missing hypotenuse of a right triangle if we know the legs of the triangle.
Math Crazy Tutoring: Pythagorean Theorem in 60 Seconds
Guidance
Teresa wants to string a clothesline across her backyard, from one corner to the opposite corner. If the yard measures 22 feet by 34 feet, how many feet of clothesline does she need?
The Pythagorean Theorem is a statement of how the lengths of the sides of a right triangle are related to each other. A right triangle is one that contains a 90 degree angle. The side of the triangle opposite the 90 degree angle is called the hypotenuse and the sides of the triangle adjacent to the 90 degree angle are called the legs.
If we let \begin{align*}a\end{align*}
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. That is: \begin{align*}a^2+b^2=c^2\end{align*}
This theorem is very useful because if we know the lengths of the legs of a right triangle, we can find the length of the hypotenuse. Also, if we know the length of the hypotenuse and the length of a leg, we can calculate the length of the missing leg of the triangle. When you use the Pythagorean Theorem, it does not matter which leg you call \begin{align*}a\end{align*}
Although nowadays we use the Pythagorean Theorem as a statement about the relationship between distances and lengths, originally the theorem made a statement about areas. If we build squares on each side of a right triangle, the Pythagorean Theorem says that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares formed by the legs of the triangle.
Use the Pythagorean Theorem and Its Converse
The Pythagorean Theorem can be used to verify that a triangle is a right triangle. If you can show that the three sides of a triangle make the equation \begin{align*}a^2+b^2=c^2\end{align*}
Note: When you use the Converse of the Pythagorean Theorem, you must make sure that you substitute the correct values for the legs and the hypotenuse. The hypotenuse must be the longest side. The other two sides are the legs, and the order in which you use them is not important.
Example A
Determine if a triangle with sides 5, 12 and 13 is a right triangle.
Solution
The triangle is right if its sides satisfy the Pythagorean Theorem.
If it is a right triangle, the longest side has to be the hypotenuse, so we let \begin{align*}c = 13\end{align*}
We then designate the shorter sides as \begin{align*}a = 5\end{align*}
We plug these values into the Pythagorean Theorem:
\begin{align*}a^2+b^2 = c^2 & \Rightarrow 5^2+12^2=c^2\\
25+144=169 = c^2 & \Rightarrow c=13\end{align*}
Example B
Determine if a triangle with sides, \begin{align*}\sqrt{10}, \sqrt{15}\end{align*}
Solution
The longest side has to be the hypotenuse, so \begin{align*}c = 5\end{align*}
We designate the shorter sides as \begin{align*}a = \sqrt{10}\end{align*}
We plug these values into the Pythagorean Theorem:
\begin{align*}a^2+b^2 = c^2 & \Rightarrow \left(\sqrt{10}\right)^2+\left(\sqrt{15}\right)^2=c^2\\
10+15=25 = c^2 & \Rightarrow c=5\end{align*}
Example C
In a right triangle one leg has length 4 and the other has length 3. Find the length of the hypotenuse.
Solution
\begin{align*}\text{Start with the Pythagorean Theorem:} && a^2+b^2& =c^2\\
\text{Plug in the known values of the legs:} && 3^2+4^2& =c^2\\
\text{Simplify:} && 9+16& =c^2\\
&& 25& =c^2\\
\text{Take the square root of both sides:} && c& =5\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: The Pythagorean Theorem and Its Converse
Vocabulary
 The Pythagorean Theorem is a statement of how the lengths of the sides of a right triangle are related to each other. A right triangle is one that contains a 90 degree angle. The side of the triangle opposite the 90 degree angle is called the hypotenuse and the sides of the triangle adjacent to the 90 degree angle are called the legs.
 If we let \begin{align*}a\end{align*}
a and \begin{align*}b\end{align*}b represent the legs of the right triangle and \begin{align*}c\end{align*}c represent the hypotenuse then the Pythagorean Theorem can be stated as:
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. That is: \begin{align*}a^2+b^2=c^2\end{align*}
Guided Practice
Determine if a triangle with sides, \begin{align*}2, \sqrt{21}\end{align*}
Solution
The longest side has to be the hypotenuse, so \begin{align*}c = 5\end{align*}
We designate the shorter sides as \begin{align*}a =2\end{align*}
We plug these values into the Pythagorean Theorem:
\begin{align*}a^2+b^2 = c^2 & \Rightarrow \left(2\right)^2+\left(\sqrt{21}\right)^2=c^2\\
4+21=25 = c^2 & \Rightarrow c=5\end{align*}
The sides of the triangle satisfy the Pythagorean Theorem, thus the triangle is a right triangle.
Practice
Determine whether each set of three numbers could be the side lengths of a right triangle.

\begin{align*}a = 12, b = 9, c = 15\end{align*}
a=12,b=9,c=15 
\begin{align*}a = 6, b = 6, c = 6 \sqrt{2}\end{align*}
a=6,b=6,c=62√ 
\begin{align*}a = 8, b =8 \sqrt{3}, c = 16\end{align*}
a=8,b=83√,c=16 
\begin{align*}a =2 \sqrt{14}, b = 5, c = 9\end{align*}
a=214−−√,b=5,c=9 
\begin{align*}a = 13, b = 16, c = 19\end{align*}
a=13,b=16,c=19 
\begin{align*}a = 20, b = 99, c = 101\end{align*}
a=20,b=99,c=101 
\begin{align*}a = 21, b = 220, c = 221\end{align*}
a=21,b=220,c=221 
\begin{align*}a = 7, b = 2, c = \sqrt{50}\end{align*}
a=7,b=2,c=50−−√ 
\begin{align*}a =8, b = 6, c = 10\end{align*}
a=8,b=6,c=10 
\begin{align*}a = 7, b =\sqrt{404}, c = 25\end{align*}
a=7,b=404−−−√,c=25
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converse
If a conditional statement is (if , then ), then the converse is (if , then . Note that the converse of a statement is not true just because the original statement is true.Hypotenuse
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.Legs of a Right Triangle
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.Pythagorean Theorem
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by , where and are legs of the triangle and is the hypotenuse of the triangle.Image Attributions
Here you'll learn how to use the Pythagorean Theorem to determine if three side lengths make a right triangle.