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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• distance formula
• slope of parallel and perpendicular lines
• rectangular coordinates
• different forms of the straight line
• solving simultaneous equations

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 .

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).

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.

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.

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

Line RS has equation

Line FG intersects with line RS when

Solving this gives us

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

Point S is the intersection of the lines 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,

is

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:

Example 1

Find the perpendicular distance from the point (5, 6) to the line -2x + 3y + 4 = 0, using the formula we just found.

Answer

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Example 2

Find the distance from the point (-3, 7) to the line

Answer