### Distance Formula in the Coordinate Plane

The distance between two points \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2)\end{align*} can be defined as \begin{align*}d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\end{align*}. This is called the **distance formula**. Remember that distances are always positive!

What if you were given the coordinates of two points? How could you find how far apart these two points are?

### Examples

#### Example 1

Find the distance between (-2, -3) and (3, 9).

Use the distance formula, plug in the points, and simplify.

\begin{align*}d & = \sqrt{(3-(-2))^2 + (9-(-3))^2}\\ & = \sqrt{(5)^2 + (12)^2}\\ & = \sqrt{25+144}\\ & = \sqrt{169} = 13 \ units\end{align*}

#### Example 2

Find the distance between (12, 26) and (8, 7).

Use the distance formula, plug in the points, and simplify.

\begin{align*}d & = \sqrt{(8-12)^2 + (7-26)^2}\\ & = \sqrt{(-4)^2 + (-19)^2}\\ & = \sqrt{16+361}\\ & = \sqrt{377} \approx 19.42 \ units\end{align*}

#### Example 3

Find the distance between (4, -2) and (-10, 3).

Plug in (4, -2) for \begin{align*}(x_1, y_1)\end{align*} and (-10, 3) for \begin{align*}(x_2, y_2)\end{align*} and simplify.

\begin{align*}d& = \sqrt{(-10-4)^2+(3+2)^2}\\ & = \sqrt{(-14)^2 + (5)^2}\\ & = \sqrt{196+25}\\ & = \sqrt{221} \approx 14.87 \ units\end{align*}

#### Example 4

Find the distance between (3, 4) and (-1, 3).

Plug in (3, 4) for \begin{align*}(x_1, y_1)\end{align*} and (-1, 3) for \begin{align*}(x_2, y_2)\end{align*} and simplify.

\begin{align*}d& = \sqrt{(-1-3)^2+(3-4)^2}\\ & = \sqrt{(-4)^2 + (-1)^2}\\ & = \sqrt{16+1}\\ & = \sqrt{17} \approx 4.12 \ units\end{align*}

#### Example 5

Find the distance between (4, 23) and (8, 14).

Plug in (4, 23) for \begin{align*}(x_1, y_1)\end{align*} and (8, 14) for \begin{align*}(x_2, y_2)\end{align*} and simplify.

\begin{align*}d& = \sqrt{(8-4)^2+(14-23)^2}\\ & = \sqrt{(4)^2 + (-9)^2}\\ & = \sqrt{16+81}\\ & = \sqrt{97} \approx 9.85 \ units\end{align*}

### Review

Find the distance between each pair of points. Round your answer to the nearest hundredth.

- (4, 15) and (-2, -1)
- (-6, 1) and (9, -11)
- (0, 12) and (-3, 8)
- (-8, 19) and (3, 5)
- (3, -25) and (-10, -7)
- (-1, 2) and (8, -9)
- (5, -2) and (1, 3)
- (-30, 6) and (-23, 0)
- (2, -2) and (2, 5)
- (-9, -4) and (1, -1)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.10.