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All courses Geometry · G-GPE.7 13 of 57
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.

Find the distance between two points on the coordinate plane

Problem
Find the distance between the points \((0,~0)\) and \((3,~4)\).
Actual coordinate polygon or dimensioned source figure containing only information supplied in the prompt. Open full size
Your answer
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Hint

Find the horizontal and vertical changes, then use the distance formula.

Distance between two points is the square root of the sum of the squared coordinate changes.

Solution walkthrough

01

Find coordinate changes

\[\text{change}~\text{in}~x=3-0=3;~\text{change}~\text{in}~y=4-0=4\]

The given points differ by 3 horizontally and 4 vertically.

02

Apply the distance formula

\[\text{distance}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}\]

The formula combines the perpendicular coordinate changes as squared legs.

03

Take the nonnegative square root

\[\sqrt{25}=5\]

Distance is nonnegative, so the requested distance is 5.

04

State the answer

\[5\]

The two points are 5 coordinate units apart.

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Another way

  1. Recognize a 3-4-5 right triangle formed by the coordinate changes.

!

Common mistake

Do not add 3+4. Straight-line distance is the hypotenuse, not the sum of horizontal and vertical travel.

Annotated polygon diagram showing dimensions, decomposition, coordinate-area products, or both absolute-value configurations and the result.
distance: 5

Solution walkthrough video