4.40: Applications of the Distance Formula

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.

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

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.