Perpendicular Distance from a Point to a Line

(BTW - we don't really need to say 'perpendicular' because the distance from a point to a line always means the shortest distance.)

This is a great problem because it uses all these things that we have learned so far:

Later, on this page...

Example using perpendicular distance formula

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The distance from a point (m, n) to the line Ax + By + C = 0 is given by:

There are some examples using this formula following the proof.

Proof of the Perpendicular Distance Formula

Let's start with the line Ax + By + C = 0 and label it DE. It has slope .

math expression

We have a point P with coordinates (m, n). We wish to find the perpendicular distance from the point P to the line (that is, distance PQ).

math expression

We now do a trick to make things easier for ourselves (the algebra is really horrible otherwise). We construct a line parallel to DE through (m, n). This line will also have slope , since it is parallel to DE. We will call this line FG.

math expression

Now we construct another line parallel to PQ passing through the origin.

This line will have slope , because it is perpendicular to DE.

Let's call it line RS. We extend it to the origin (0, 0).

We will find the distance RS, which I hope you agree is equal to the distance PQ that we wanted at the start.

math expression

Since FG passes through (m, n) and has slope , its equation is math expression or .

Line RS has equation

Line FG intersects with line RS when

Solving this gives us

So after substituting this back into math expression , we find that point R is

Point S is the intersection of the lines math expression and Ax + By + C = 0 (which can be written ).

This occurs when (that is, we are solving them simultaneously)

Solving for x gives

Finding y by substituting back into

gives

So S is the point

So the distance RS, using the distance formula,

math expression

The absolute value sign is necessary since distance must be a positive value, and certain combinations of A, m , B, n and C can produce a negative number in the numerator.

So the distance from the point (m, n) to the line Ax + By + C = 0 is given by: