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The Distance Formula Solutions

• The distance between any two points on a number line is calculated as $| a–b |$ or $| b–a |$ because distance    must    be a    positive    value. 

• The    distance formula    is used to determine the distance between two    ordered pairs    on the coordinate plane. 

• The distance formula is: $d=\sqrt{{\left(x_2– x_1\right)}^2+{\left(y_2– y_1\right)}^2}$
• It uses $\underline{ \left(x_1, y_1\right), \left(x_2, y_2\right) }$ and is a derivation of the Pythagorean Theorem.
• The distance formula allows you to solve some problems without having to    graph    them on the coordinate plane.

Note

Read the problem carefully to determine if a graph is needed

• Distance, or length, is not typically written in radical form because that is not how they are measured. Instead, enter the entire formula into a    calculator   .

• The result is likely a    rounded value    because most numbers are not perfect squares.

Note

The distance formula is often used in conjunction with other equations and formulas to solve problems. You can also write solutions to distance formula problems in simplified radical form. To review simplifying radicals, refer to the lessons in Unit 2.

Example 1

Plot and connect the ordered pairs. Then find the perimeter of the figure. Round to the nearest hundredth.

$\left\{P(0, 0), Q(2.5, 4), R(4, 1.5)\right\}$

Note

Because none of the sides of the triangle are on a horizontal or vertical line on the coordinate plane, the distance formula is the best way to find the length of each side.

Individual line segment distances are shown with one additional place value than the instructions require so that you can confirm the rounded answer. You can skip this step if you choose.

Example 2

Find the distance between the vertex and the y-intercept of the parabola. Round to the nearest hundredth.

$y={(x+5)}^2–8$

$$\begin{gathered} d=\sqrt{{\left(–5–0\right)}^2+{\left(–8–17\right)}^2}=25.495 \\ d= 25.50 \mathrm{units} \end{gathered}$$