While painting a wall in your home one day, you realize that the wall you are painting seems "tilted", as though it might fall over. You realize that if the wall is standing upright, the angle between the wall and the floor is ninety degrees. After a few careful measurements, you find that the distance from the bottom of the ladder to the wall is 3 feet, the top of the ladder is at a point 10 feet up on the wall, and the ladder is 12 feet long. Can you determine if the wall is still standing upright, or if it is starting to lean?

### Classifying Triangles by Using the Pythagorean Theorem

We can use the Pythagorean Theorem to help determine if a triangle is a right triangle, if it is acute, or if it is obtuse.

To help you visualize this, think of an equilateral triangle with sides of length \begin{align*}5\end{align*}. We know that this is an acute triangle. If you plug in \begin{align*}5\end{align*} for each number in the Pythagorean Theorem we get \begin{align*}5^2 + 5^2 = 5^2\end{align*} and \begin{align*}50 > 25\end{align*}. Therefore, if \begin{align*}a^2 + b^2 > c^2\end{align*}, then lengths \begin{align*}a, \ b\end{align*}, and \begin{align*}c\end{align*} make up an acute triangle. Conversely, if \begin{align*}a^2 + b^2 < c^2\end{align*}, then lengths \begin{align*}a, \ b\end{align*}, and \begin{align*}c\end{align*} make up the sides of an obtuse triangle. *It is important to note that the length \begin{align*}``c''\end{align*} is always the longest.*

#### Using the Pythagorean Theorem

Determine if the following lengths make an acute, right or obtuse triangle.

1. 5, 6, 7

Plug in each set of lengths into the Pythagorean Theorem.

\begin{align*}5^2 + 6^2 & \ ? \ 7^2\\ 25 + 36 & \ ? \ 49\\ 61 & > 49\end{align*}

Because \begin{align*}61>49\end{align*}, this is an acute triangle.

2. 5, 10, 14

Plug in each set of lengths into the Pythagorean Theorem.

\begin{align*}5^2 + 10^2 & \ ? \ 14^2\\ 25+ 100 & \ ? \ 196\\ 125 & < 196\end{align*}

Because \begin{align*}125<196\end{align*}, this is an obtuse triangle.

3. 12, 35, 37

Plug in each set of lengths into the Pythagorean Theorem.

\begin{align*}12^2 + 35^2 & \ ? \ 37^2\\ 144 + 1225 & \ ? \ 1369\\ 1369 & = 1369\end{align*}

Because the two sides are equal, this is a right triangle.

NOTE: All of the lengths in the above problem represent the lengths of the sides of a triangle. Recall the Triangle Inequality Theorem from geometry which states: The length of a side in a triangle is less than the sum of the other two sides. For example, 4, 7 and 13 cannot be the sides of a triangle because \begin{align*}4+7\end{align*} is not greater than 13.

### Examples

#### Example 1

Earlier, you were given a problem asking if the wall is still standing upright, or if it is starting to lean.

The ladder is making a triangle with the floor as one side, the wall as another, and the ladder itself serves as the hypotenuse. To see if the wall is leaning, you can determine the type of triangle that is made with these lengths (right, acute, or obtuse). If the triangle is a right triangle, then the wall is standing upright. Otherwise, it is leaning.

Plugging the lengths of the sides into the Pythagorean Theorem:

\begin{align*}3^2 + 10^2 & \ ? \ 12^2\\ 9 + 100 & \ ? \ 144\\ 109 & < 144\end{align*}

Yes, you were right. Because 109 < 144, this is an obtuse triangle. The wall is leaning with an angle greater than ninety degrees.

#### Example 2

Determine if the following lengths make an acute, right or obtuse triangle.

8, 15, 20

Plug in each set of lengths into the Pythagorean Theorem.

\begin{align*}8^2 + 15^2 & \ ? \ 20^2\\ 64 + 225 & \ ? \ 400\\ 289 & < 400\end{align*}

Because \begin{align*}289<400\end{align*}, this is an obtuse triangle.

#### Example 3

Determine if the following lengths make an acute, right or obtuse triangle.

15, 22, 25

Plug in each set of lengths into the Pythagorean Theorem.

\begin{align*}15^2 + 22^2 & \ ? \ 25^2\\ 225 + 484 & \ ? \ 625\\ 709 & > 625\end{align*}

Because \begin{align*}709>625\end{align*}, this is an acute triangle.

### Review

Determine if each of the following lengths make a right triangle.

- 9, 40, 41.
- 12, 24, 26.
- 5, 10, 14.
- 3, \begin{align*}3\sqrt{3}\end{align*}, 6.

Determine if the following lengths make an acute, right or obtuse triangle.

- 10, 15, 18.
- 4, 20, 21.
- 15, 16, 17.
- 15, 15, \begin{align*}15\sqrt{2}\end{align*}.
- 12, 17, 19.
- 3, 4, 5.
- 12, \begin{align*}12\sqrt{3}\end{align*}, 24.
- 2, 4, 5.
- 3, 5, 7.
- Explain why if \begin{align*}a^2 + b^2 < c^2\end{align*} then the triangle is obtuse.
- Explain why if \begin{align*}a^2 + b^2 > c^2\end{align*} then the triangle is acute.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.3.