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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CK-12 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*} and \begin{align*}b\end{align*} represent the legs of the right triangle and \begin{align*}c\end{align*} 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*}.

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*} and which leg you call \begin{align*}b\end{align*}, but the hypotenuse is always called \begin{align*}c\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*} true, then you know that the triangle is a right triangle. This is called the **Converse of the Pythagorean Theorem.**

**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*} and \begin{align*}b = 12\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*}

The sides of the triangle satisfy the Pythagorean Theorem, thus **the triangle is a right triangle.**

#### Example B

*Determine if a triangle with sides, \begin{align*}\sqrt{10}, \sqrt{15}\end{align*} and 5 is a right triangle.*

**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*} and \begin{align*}b = \sqrt{15}\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*}

The sides of the triangle satisfy the Pythagorean Theorem, thus **the triangle is a right triangle.**

#### 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.

CK-12 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*} and \begin{align*}b\end{align*} represent the legs of the right triangle and \begin{align*}c\end{align*} 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*} and 5 is a right triangle.*

**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*} and \begin{align*}b = \sqrt{21}\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*}
- \begin{align*}a = 6, b = 6, c = 6 \sqrt{2}\end{align*}
- \begin{align*}a = 8, b =8 \sqrt{3}, c = 16\end{align*}
- \begin{align*}a =2 \sqrt{14}, b = 5, c = 9\end{align*}
- \begin{align*}a = 13, b = 16, c = 19\end{align*}
- \begin{align*}a = 20, b = 99, c = 101\end{align*}
- \begin{align*}a = 21, b = 220, c = 221\end{align*}
- \begin{align*}a = 7, b = 2, c = \sqrt{50}\end{align*}
- \begin{align*}a =8, b = 6, c = 10\end{align*}
- \begin{align*}a = 7, b =\sqrt{404}, c = 25\end{align*}