# Applications of the Pythagorean Theorem

## Height, distance, and angles or types of triangles.

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Applications of the Pythagorean Theorem

### Applications of the Pythagorean Theorem

#### Find the Height of an Isosceles Triangle

One way to use The Pythagorean Theorem is to find the height of an isosceles triangle (see Example 1).

#### Prove the Distance Formula

Another application of the Pythagorean Theorem is the Distance Formula. We will prove it here.

Let’s start with point \begin{align*}A(x_1, y_1)\end{align*} and point \begin{align*}B(x_2, y_2)\end{align*}. We will call the distance between \begin{align*}A\end{align*} and \begin{align*}B, d\end{align*}.

Draw the vertical and horizontal lengths to make a right triangle.

Now that we have a right triangle, we can use the Pythagorean Theorem to find the hypotenuse, \begin{align*}d\end{align*}.

\begin{align*}d^2 &= (x_1-x_2)^2 + (y_1-y_2)^2\\ d &= \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*}

Distance Formula: The distance between \begin{align*}A(x_1, y_1)\end{align*} and \begin{align*}B(x_2, y_2)\end{align*} is \begin{align*}d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*}.

#### Classify a Triangle as Acute, Right, or Obtuse

We can extend the converse of the Pythagorean Theorem to determine if a triangle is an obtuse or acute triangle.

Acute Triangles: If the sum of the squares of the two shorter sides in a right triangle is greater than the square of the longest side, then the triangle is acute.

For \begin{align*}b < c\end{align*} and \begin{align*}a < c\end{align*}, if \begin{align*}a^2 + b^2 > c^2\end{align*}, then the triangle is acute.

Obtuse Triangles: If the sum of the squares of the two shorter sides in a right triangle is less than the square of the longest side, then the triangle is obtuse.

For \begin{align*}b < c\end{align*} and \begin{align*}a < c\end{align*}, if \begin{align*}a^2+b^2, then the triangle is obtuse.

What if you were given an equilateral triangle in which all the sides measured 4 inches? How could you use the Pythagorean Theorem to find the triangle's altitude?

### Examples

#### Example 1

What is the height of the isosceles triangle?

Draw the altitude from the vertex between the congruent sides, which will bisect the base.

\begin{align*}7^2 + h^2 &= 9^2\\ 49 + h^2 &= 81\\ h^2 &= 32\\ h &= \sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2}\end{align*}

#### Example 2

Find the distance between (1, 5) and (5, 2).

Make \begin{align*}A(1, 5)\end{align*} and \begin{align*}B(5, 2)\end{align*}. Plug into the distance formula.

\begin{align*}d &= \sqrt{(1-5)^2 + (5-2)^2}\\ &= \sqrt{(-4)^2 + (3)^2}\\ &= \sqrt{16+9} = \sqrt{25} = 5\end{align*}

Just like the lengths of the sides of a triangle, distances are always positive.

#### Example 3

Graph \begin{align*}A(-4, 1), B(3, 8)\end{align*}, and \begin{align*}C(9, 6)\end{align*}. Determine if \begin{align*}\triangle ABC\end{align*} is acute, obtuse, or right.

Use the distance formula to find the length of each side.

\begin{align*}AB &= \sqrt{(-4-3)^2 + (1-8)^2} = \sqrt{49+49} = \sqrt{98}\\ BC &= \sqrt{(3-9)^2 + (8-6)^2} = \sqrt{36 + 4} = \sqrt{40}\\ AC &= \sqrt{(-4-9)^2 + (1-6)^2} = \sqrt{169 + 25} = \sqrt{194}\end{align*}

Plug these lengths into the Pythagorean Theorem.

\begin{align*}\left( \sqrt{98} \right )^2 + \left( \sqrt{40} \right)^2 & \ ? \ \left ( \sqrt{194} \right )^2\\ 98 + 40 & \ ? \ 194\\ 138 & < 194\end{align*}

\begin{align*}\triangle ABC\end{align*} is an obtuse triangle.

For Examples 4 and 5, determine if the triangles are acute, right or obtuse.

#### Example 4

Set the longest side to \begin{align*}c\end{align*}.

\begin{align*}15^2 + 14^2 & \ ? \ 21^2\\ 225 + 196 & \ ? \ 441\\ 421 & < 441\end{align*}

The triangle is obtuse.

#### Example 5

Set the longest side to \begin{align*}c\end{align*}.

A triangle with side lengths 5, 12, 13.

\begin{align*}5^2 +12^2 = 13^2\end{align*} so this triangle is right.

### Review

Find the height of each isosceles triangle below. Simplify all radicals.

Find the length between each pair of points.

1. (-1, 6) and (7, 2)
2. (10, -3) and (-12, -6)
3. (1, 3) and (-8, 16)
4. What are the length and width of a 42” HDTV? Round your answer to the nearest tenth.
5. Standard definition TVs have a length and width ratio of 4:3. What are the length and width of a 42” Standard definition TV? Round your answer to the nearest tenth.

Determine whether the following triangles are acute, right or obtuse.

1. 7, 8, 9
2. 14, 48, 50
3. 5, 12, 15
4. 13, 84, 85
5. 20, 20, 24
6. 35, 40, 51
7. 39, 80, 89
8. 20, 21, 38
9. 48, 55, 76

Graph each set of points and determine whether \begin{align*}\triangle ABC\end{align*} is acute, right, or obtuse, using the distance formula.

1. \begin{align*}A(3, -5), B(-5, -8), C(-2, 7)\end{align*}
2. \begin{align*}A(5, 3), B(2, -7), C(-1, 5)\end{align*}
3. \begin{align*}A(1, 6) , B(5, 2), C(-2, 3)\end{align*}
4. \begin{align*}A(-6, 1), B(-4, -5), C(5, -2)\end{align*}

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### Vocabulary Language: English Spanish

TermDefinition
acute triangle A triangle where all angles are less than $90^\circ$.
Obtuse Triangle An obtuse triangle is a triangle with one angle that is greater than 90 degrees.
Distance Formula The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be defined as $d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.
Vertex A vertex is a point of intersection of the lines or rays that form an angle.