# 4.40: Applications of the Distance Formula

- Page ID
- 4978

Length between two points based on a right triangle.

## Distance Formula in the Coordinate Plane

The distance between two points \((x_1, y_1)\) and \((x_2,y_2)\) can be defined as \(d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}\). 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?

Example \(\PageIndex{1}\)

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

**Solution**

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

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

Example \(\PageIndex{2}\)

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

**Solution**

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

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

Example \(\PageIndex{3}\)

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

**Solution**

Plug in \((4, -2)\) for \((x_1, y_1)\) and \((-10, 3)\) for \((x_2,y_2)\) and simplify.

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

Example \(\PageIndex{4}\)

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

**Solution**

Plug in (3, 4)\) for \((x_1, y_1)\) and \((-1, 3)\) for \((x_2,y_2)\) and simplify.

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

Example \(\PageIndex{5}\)

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

**Solution**

Plug in \((4, 23)\) for \((x_1, y_1)\) and \((8, 14)\) for \((x_2,y_2)\) and simplify.

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

## 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.

## Resource

## Vocabulary

Term | Definition |
---|---|

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. |

## Additional Resources

Interactive Element

Video: The Distance Formula

Activities: Distance Formula in the Coordinate Plane Discussion Questions

Study Aids: Segments Study Guide

Practice: Applications of the Distance Formula

Real World: Distance Formula in the Coordinate Plane