Last year, Yuli and his dad made a square bed for vegetables and edged it with timbers. This year his mom wants another one for flowers that measures 5 feet by 5 feet and has a timber diagonally through the middle to separate the sections. How long should the dividing timber be?

In this concept, you will learn how to use the Pythagorean Theorem and its converse.

### Using the Pythagorean Theorem and its Converse

You already know that a square consists of four equal sides and four equal, right, angles.

When you divide a square in half by a diagonal, it forms two right triangles.

In the labeled right triangle below:

Sides \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are adjacent to the right angle and are called the **legs** of the triangle.

Side \begin{align*}c\end{align*}, the longest side, is opposite the right angle and is called the **hypotenuse** of the triangle.

The **Pythagorean Theorem** is named for the Greek mathematician Pythagoras, who discovered the unique relationship between the dimensions of a right triangle. The theorem says that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

\begin{align*}a^2 + b^2 = c^2\end{align*}

Let’s look at an example.

Find the hypotenuse.

First, write the formula, and substitute what you know.

\begin{align*}\begin{array}{rcl} a^2 + b^2 &=& c^2 \\ 3^2 + 4^2 &=& c^2 \end{array}\end{align*}

Next, do the calculations.

\begin{align*}\begin{array}{rcl} 9 + 16 &=& c^2 \\ 25 &=& c^2 \end{array}\end{align*}

Then, take the square root of both sides of the equation.

\begin{align*}5 = c\end{align*}

The answer is 5. The hypotenuse of the triangle is 5.

Here’s another example.

Find the length of the hypotenuse in the triangle below by using the Pythagorean Theorem.

First, write the formula, and substitute.

\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=& 6^2+8^2 \end{array}\end{align*}

Next, calculate the squares.

\begin{align*}\begin{array}{rcl} c^2&=&36+64 \\ c^2&=& 100 \end{array}\end{align*}

Then take the square root of each side.

The answer is \begin{align*}c=10\end{align*}. The length of the hypotenuse is 10.

### Examples

#### Example 1

Earlier, you were given a problem about Yuli and his garden beds.

Yuli needs to know how long a diagonal timber he needs for a \begin{align*}5 \ ft \times 5 \ ft\end{align*} square.

First, use the Pythagorean Theorem and substitute in the values you know.

\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=&5^2+5^2 \end{array}\end{align*}

Next, calculate the value of the squares.

\begin{align*}\begin{array}{rcl} c^2 &=&25+25 \\ c^2&=&50 \end{array}\end{align*}

Then take the square root of each side.

\begin{align*}\sqrt{c^2}=\sqrt{50}\end{align*}

Use your calculator.

\begin{align*}c = 7.071067812 \ldots\end{align*}

Round the number.

\begin{align*}c\approx 7.07\end{align*}

The answer is that the hypotenuse \begin{align*}c \approx 7.07\end{align*}, but since Yuli may have a hard time measuring \begin{align*}\frac{7}{100}\end{align*} of a foot, it is safe to say that Yuli can use a 7 ft timber.

#### Example 2

What is the length of the hypotenuse of this triangle?

First, write the formula and fill in what you know.

\begin{align*}7^2 + 10^2 = c^2\end{align*}

Next, do the calculations.

\begin{align*}\begin{array}{rcl} 49 + 100 &=& c^2 \\ 149 &=& c^2 \end{array}\end{align*}

Then take the square roots.

\begin{align*}\sqrt{149}=c\end{align*}

Since 149 is not a perfect square, use your calculator.

\begin{align*}\sqrt{149} = 12.20655562 \ldots\end{align*}

Round to the nearest tenth.

The answer is \begin{align*}\sqrt{149}=12.2\end{align*}

#### Example 3

A right triangle has legs that measure 9 and 12.

What is the hypotenuse?

First, fill the values into the equation.

\begin{align*}9^2+12^2=c^2\end{align*}

Next, perform the calculations.

\begin{align*}\begin{array}{rcl} 81+144&=& c^2 \\ 225&=& c^2 \end{array}\end{align*}

Then, determine the square roots.

\begin{align*}15 = c\end{align*}

The answer is 15. The hypotenuse is 15.

#### Example 4

Find the hypotenuse of a right triangle with legs of 3 and 6.

\begin{align*}\begin{array}{rcl} a^2+b^2=c^2 \\ 3^2+6^2=c^2 \end{array}\end{align*}

Next, square the legs.

\begin{align*}\begin{array}{rcl} 9+36=c^2 \\ 45=c^2 \end{array}\end{align*}

Then calculate the square roots of both sides of the equation.

\begin{align*}\sqrt{45}=c\end{align*}

Use a calculator.

\begin{align*}c = 6.708203933 \ldots\end{align*}

The answer is \begin{align*}c \approx 6.7\end{align*}. The hypotenuse is approximately 6.7.

#### Example 5

Claire’s mom needs to fix a broken shingle on their house. If the shingle is 16 feet above the ground and Claire places the foot of the ladder 12 feet from the wall, how long will the ladder need to be?

First, draw a picture to help determine what you are solving for.

In this picture, you can see that the ladder is the hypotenuse.

Next, insert the values of the legs into the formula for the Pythagorean Theorem.

\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=& 12^2+16^2 \end{array}\end{align*}

Next, calculate the value of the squares.

\begin{align*}\begin{array}{rcl} c^2&=& 144+256 \\ c^2&=&400 \end{array}\end{align*}

Then take the square root of each side.

The answer is \begin{align*}c=20\end{align*}. The length of the hypotenuse is 10, and the ladder needs to be 10 feet long.

### Review

Use the Pythagorean Theorem to find the length of side \begin{align*}c\end{align*}, the hypotenuse. You may round to the nearest hundredth when necessary.

- \begin{align*}a = 6, b = 8, c =?\end{align*}
- \begin{align*}a = 9, b = 12, c =?\end{align*}
- \begin{align*}a = 12, b = 16, c =?\end{align*}
- \begin{align*}a = 15, b = 20, c =?\end{align*}
- \begin{align*}a = 18, b = 24, c =?\end{align*}
- \begin{align*}a = 24, b = 32, c =?\end{align*}
- \begin{align*}a = 5, b = 7, c =?\end{align*}
- \begin{align*}a = 9, b = 11, c =?\end{align*}
- \begin{align*}a = 10, b = 12, c =?\end{align*}
- \begin{align*}a = 12, b = 8, c =?\end{align*}
- \begin{align*}a = 11, b = 8, c =?\end{align*}
- \begin{align*}a = 9, b = 8, c =?\end{align*}
- \begin{align*}a = 15, b = 7, c =?\end{align*}
- \begin{align*}a = 14, b = 3, c =?\end{align*}
- \begin{align*}a = 12, b = 8, c =?\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.7.