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:

• 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:

`d=(|Am+Bn+C|)/(sqrt(A^2+B^2`

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 `-A/B`.

We have a point P with coordinates (m, n). We wish to find the perpendicular distance from the point P to the line DE (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 `-A/B`, 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 `B/A`, 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 `-A/B`, its equation is `y-n=-A/B(x-m)` or

`y=(-Ax+Am+Bn)/B`.

Line RS has equation `y=B/Ax.`

Line FG intersects with line RS when

`B/Ax=(-Ax+Am+Bn)/B`

Solving this gives us

`x=(A(Am+Bn))/(A^2+B^2`

So after substituting this back into `y=B/Ax,` we find that point R is

`((A(Am+Bn))/(A^2+B^2),(B(Am+Bn))/(A^2+B^2))`

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Point S is the intersection of the lines `y=B/Ax` and Ax + By + C = 0, which can be written `y=-(Ax+C)/B`.

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

`-(Ax+C)/B=B/Ax`

Solving for x gives

`x=(-AC)/(A^2+B^2)`

Finding y by substituting back into

`y=B/Ax`

gives

`y=B/A((-AC)/(A^2+B^2))=(-BC)/(A^2+B^2`

So S is the point

`((-AC)/(A^2+B^2),(-BC)/(A^2+B^2))`

The distance RS, using the distance formula,

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

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:

`d=(|Am+Bn+C|)/(sqrt(A^2+B^2`

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

Example 2

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

`y=6/5x+2`

Answer

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