Points and Distance

Distance Formula

Find the exact distance between two coordinate points by using the horizontal and vertical changes as the legs of a right triangle.

Example

Plot two points, then compute the exact distance algebraically.

highlighted = computed this step

Step 1 — Plot points

Plot points A and B with the segment.

Coordinate plot\text{Coordinate plot}
Distance formula plotSegment from A(1, 2) to B(5, 4).AB

Step 2 — Coordinates

Read the coordinates of A and B.

A=(1,2)B=(5,4)A=( \hl{1} , \hl{2} ) \quad B=( \hl{5} , \hl{4} )

Step 3 — Differences

Subtract coordinates to find the changes.

Δx=5−1=4Δy=4−2=2\Delta x= 5 - 1 = \hl{4} \quad \Delta y= 4 - 2 = \hl{2}

Step 4 — Square and add

Square the changes and add.

d=42+22=20d=\sqrt{ 4 ^{2} + 2 ^{2} }=\sqrt{ \hl{20} }

Step 5 — Distance

Simplify the exact distance.

d=20=25d=\sqrt{ 20 }= 2\sqrt{5}
distance-formula d = √((x2-x1)^2 + (y2-y1)^2)