Lines

Distance from Point to Line

Compute the shortest distance from a point to a line using the exact standard-form distance formula.

Example

Plot the shortest distance and compute it exactly.

highlighted = computed this step

Step 1 — Plot point and line

Plot the point, line, and perpendicular distance.

Coordinate plot\text{Coordinate plot}
Point to line distance plotPoint P(3, 4) and the perpendicular foot F(6/5, 2/5) on the line.linePF

Step 2 — Point and line

Read the point and standard-form line.

P=(3,4)1x+2y−2=0P=( \hl{3} , \hl{4} ) \quad \hl{1} x + \hl{2} y - \hl{2} = 0

Step 3 — Numerator

Substitute the point into the numerator.

∣1⋅3+2⋅4−2∣=9| 1 \cdot 3 + 2 \cdot 4 - 2 |= \hl{9}

Step 4 — Denominator

Compute the square-root denominator.

12+22=5\sqrt{ 1 ^{2} + 2 ^{2} }=\sqrt{ \hl{5} }

Step 5 — Exact distance

Write the exact point-to-line distance.

d=9/5=95/5d= 9 /\sqrt{ 5 }= 9\sqrt{5} / \hl{5}
distance-point-to-line d = |ax0 + by0 + c| / √(a^2 + b^2)