What if a friend of yours wanted to design a rectangular building with one wall 65 ft long and the other wall 72 ft long? How can he ensure the walls are going to be perpendicular?

### Pythagorean Theorem and Pythagorean Triples

The sides of a right triangle are called legs (the sides of the right angle) and the side opposite the right angle is the hypotenuse. For the Pythagorean Theorem, the legs are “\begin{align*}a\end{align*}” and “\begin{align*}b\end{align*}” and the hypotenuse is “\begin{align*}c\end{align*}”.

**Pythagorean Theorem:** Given a right triangle with legs of lengths \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and a hypotenuse of length \begin{align*}c\end{align*}, then \begin{align*}a^2 + b^2 = c^2\end{align*}.

**Pythagorean Theorem Converse:** If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

There are several proofs of the Pythagorean Theorem, shown below.

#### Investigation: Proof of the Pythagorean Theorem

Tools Needed: pencil, 2 pieces of graph paper, ruler, scissors, colored pencils (optional)

- On the graph paper, draw a 3 in. square, a 4 in. square, a 5 in square and a right triangle with legs of 3 and 4 inches.
- Cut out the triangle and square and arrange them like the picture on the right.
- This theorem relies on area. Recall from a previous math class, that the area of a square is length times width. But, because the sides are the same you can rewrite this formula as \begin{align*}A_{square} = length \times width = side \times side = side^2\end{align*}. So, the Pythagorean Theorem can be interpreted as \begin{align*}(square \ with \ side \ a)^2 + (square \ with \ side \ b)^2 = (square \ with \ side \ c)^2\end{align*}. In this Investigation, the sides are 3, 4 and 5 inches. What is the area of each square?
- Now, we know that \begin{align*}9 + 16 = 25\end{align*}, or \begin{align*}3^2 + 4^2 = 5^2\end{align*}. Cut the smaller squares to fit into the larger square, thus proving the areas are equal.

##### Another Proof of the Pythagorean Theorem

This proof is “more formal,” meaning that we will use letters, \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} to represent the sides of the right triangle. In this particular proof, we will take four right triangles, with legs \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and hypotenuse \begin{align*}c\end{align*} and make the areas equal.

##### Pythagorean Triples

A **Pythagorean Triple** is a set of three whole numbers that makes the Pythagorean Theorem true. The most frequently used Pythagorean triple is 3, 4, 5, as in Investigation 8-1. Any multiple of a Pythagorean triple is also considered a triple because it would still be three whole numbers. Therefore, 6, 8, 10 and 9, 12, 15 are also sides of a right triangle. Other Pythagorean triples are:

\begin{align*}3, 4, 5 \qquad 5, 12, 13 \qquad 7, 24, 25 \qquad 8, 15, 17\end{align*}

There are infinitely many Pythagorean triples. To see if a set of numbers makes a triple, plug them into the Pythagorean Theorem.

#### Determining if Given Lengths Make a Right Triangle

Do 6, 7, and 8 make the sides of a right triangle?

Plug in the three numbers into the Pythagorean Theorem. ** The largest length will always be the hypotenuse**. \begin{align*}6^2 + 7^2 = 36 + 49 = 85 \neq 8^2\end{align*}. Therefore, these lengths do not make up the sides of a right triangle.

#### Finding the Length of the Hypotenuse

Find the length of the hypotenuse of the triangle below.

Let’s use the Pythagorean Theorem. Set \begin{align*}a\end{align*} and \begin{align*}b\end{align*} equal to 8 and 15 and solve for \begin{align*}c\end{align*}, the hypotenuse.

\begin{align*}8^2 + 15^2 & = c^2\\ 64 + 225 & = c^2\\ 289 & = c^2 \qquad \quad Take \ the \ square \ root \ of \ both \ sides.\\ 17 & = c\end{align*}

When you take the square root of an equation, usually the answer is +17 or -17. Because we are looking for length, we only use the positive answer. ** Length is never negative**.

#### Determining if Given Lengths are a Pythagorean Triple

Is 20, 21, 29 a Pythagorean triple?

If \begin{align*}20^2 + 21^2\end{align*} is equal to \begin{align*}29^2\end{align*}, then the set is a triple.

\begin{align*}20^2 + 21^2 & = 400 + 441 = 841\\ 29^2 & = 841\end{align*}

Therefore, 20, 21, and 29 is a Pythagorean triple.

#### Recognizing Right Triangles

Determine if the triangle below is a right triangle.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent \begin{align*}c\end{align*}, in the equation.

\begin{align*}a^2 + b^2 &= c^2\\ 8^2 + 16^2 &= \left ( 8 \sqrt{5} \right )^2\\ 64 + 256&= 64 \cdot 5\\ 320 &= 320\end{align*}

The triangle is a right triangle.

#### Earlier Problem Revisited

To make the walls perpendicular, find the length of the diagonal.

\begin{align*}65^2 + 72^2 & = c^2\\ 4225 + 5184 & = c^2\\ 9409 & = c^2\\ 97 & = c\end{align*}

In order to make the building rectangular, both diagonals must be 97 feet.

### Examples

#### Example 1

Find the missing side of the right triangle below.

Here, we are given the hypotenuse and a leg. Let’s solve for \begin{align*}b\end{align*}.

\begin{align*}7^2 + b^2 & = 14^2\\ 49 + b^2 & = 196\\ b^2 & = 147\\ b & = \sqrt{147} = \sqrt{7 \cdot 7 \cdot 3} =7 \sqrt{3}\end{align*}

#### Example 2

What is the diagonal of a rectangle with sides 10 and \begin{align*}16 \sqrt{5}\end{align*}?

For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find \begin{align*}d\end{align*}.

\begin{align*}10^2 + \left ( 16 \sqrt{5} \right )^2 & = d^2\\ 100 + 1280 & = d^2\\ 1380 & = d^2\\ d & = \sqrt{1380} = 2 \sqrt{345}\end{align*}

#### Example 3

Determine if the triangle below is a right triangle.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent \begin{align*}c\end{align*}, in the equation.

\begin{align*}a^2 + b^2 &= c^2\\ 22^2 + 24^2&= 26^2\\ 484 + 576 &= 676\\ 1060 &\neq 676\end{align*}

The triangle is not a right triangle.

### Review

Find the length of the missing side. Simplify all radicals.

- If the legs of a right triangle are 10 and 24, then the hypotenuse is _____________.
- If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
- If the legs of a right triangle are \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, then the hypotenuse is ____________.
- If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

- 12, 35, 37
- 9, 17, 18
- 10, 15, 21
- 11, 60, 61
- 15, 20, 25
- 18, 73, 75

*Pythagorean Theorem Proofs*

The first proof below is similar to the one done earlier in this Concept. Use the picture below to answer the following questions.

- Find the area of the square with sides \begin{align*}(a + b)\end{align*}.
- Find the sum of the areas of the square with sides \begin{align*}c\end{align*} and the right triangles with legs \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
- The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.

Major General James A. Garfield (and former President of the U.S) is credited with deriving this next proof of the Pythagorean Theorem using a trapezoid.

- Find the area of the trapezoid using the trapezoid area formula: \begin{align*}A = \frac{1}{2} (b_1 + b_2)h\end{align*}
- Find the sum of the areas of the three right triangles in the diagram.
- The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.1.