Chapter 4 · Geometry and Equations · Concept 2
4-12 Distance Between a Point and a Line
How far is a point from a line? There are endlessly many answers depending on where you aim — so 'the' distance means the shortest one, measured straight out at a right angle. One formula gives it.
Guiding question
How close does a beam pass to something you must not hit?
Learning outcome. Calculate the perpendicular distance from a point to a line and use it.
One formula — after one piece of tidying
the distance from (x₁, y₁) to the line ax + by + c = 0
d = |ax₁ + by₁ + c|√(a² + b²)
The formula needs the line in ax + by + c = 0 form.
Given y = 3x − 2, rearrange to 3x − y − 2 = 0 first. Given 12x − 5y = 7, move the 7 across to get 12x − 5y − 7 = 0. Putting an untidied line into the formula gives a wrong answer with no warning attached.
Two small things about the formula.
The bars on top are there because a distance is never negative — the inside can easily come out negative and that is fine. And the bottom uses a and b only: the constant c never appears there. Slipping c into the square root is a common wrong turn.
“The length of the perpendicular dropped from a point to a line” is the same thing said in more words. Both mean the shortest way out, at a right angle.
The grey routes reach the line too. The red one is the only one measured at a right angle, and it is the only one anyone calls “the distance”.
3x − 2y + 5 = 0
2. Find the length of the perpendicular OH dropped from the origin to the line 3x − y = 5.
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The line is not in the right form yet, so tidy it before anything else.
3x − y = 5 → 3x − y − 5 = 0Everything on one side, zero on the other. The formula cannot be used until this is done.
Work with a partner. Before either of you touches the formula, write the line out in ax + by + c = 0 form and read a, b, c aloud with their signs. Most wrong answers on this page are settled before any calculating happens.
Three of these arrive in the wrong form.
Questions 2, 3 and 7 are written as 12x − 5y = 7 and y = 2x − 3 and y = 3x + 1. Each needs one line of rearranging before the formula can touch it — and each gives a plausible wrong number if you skip it.
In stereotactic radiosurgery, a narrow radiation beam follows the line y = (1/3)x + 2. A critical nerve centre sits nearby at P(−1, 5). To keep it safe, find the shortest distance from the nerve centre to the beam.
Using the formula on an untidied line.
y = 3x − 2 has to become 3x − y − 2 = 0 first. Three of the seven Exercise questions are written this way on purpose.
Putting c into the square root.
The bottom is √(a² + b²) — the coefficients of x and y only. The constant never appears there.
Losing the sign of b.
In 3x − y − 5 = 0, b is −1. The sign in front of y belongs to it, and dropping it changes the top.
Dropping the bars.
The top often comes out negative. A distance cannot be, so the bars make it positive — that is not an optional tidy-up.
Leaving a root on the bottom.
5 ÷ √10 is not finished. Multiply top and bottom by √10 to get √10 ÷ 2.
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