Shortest Distance from a Point to a Line

Intuitively, it makes sense that the shortest distance between two points is a straight line. When dealing with a point and a line, the shortest distance turns out to be a perpendicular line. See the figure to the left.

It is evident from the picture that the shortest line connecting the black line and the point is the green line. It forms a right angle with the black line. With this fact established, we turn out attention to the problem of finding the shortest distance.

Suppose we are
given an arbitrary line in the form *y* = *mx* + *b* and a point
*P*(*x*_{1}, *y*_{1}). We want to find the
shortest distance between the line and the point. We shall call the point on
the line that is shortest (*x*_{0}, *y*_{0}). We have
drawn a green line between the point and line. We want to find the length of
that green line. (See the figure below.)

Notice that the
equations of both lines have been included. The first equation was given to us.
The second was found using the fact that the slope of the line perpendicular to
another line with slope *m* will be 1/*m*.
The *y*-intercept was found by plugging in the point (*x*_{1},
*y*_{1}) and solving for *b*.

Now, we want to
find the point where the two lines intersect. We know that at that point, the
two lines intersect, so we shall set them equal to each other and solve for *x*.
Then we will take that *x*-value and plug it into the first line, thereby
solving for *y*. That will give us the point (*x*_{0}, *y*_{0}).

Setting the two lines equal to each other, we have . Adding and subtracting terms, we get that:

Now that we know *x*_{0},
we can find plug that into the line *y* = *mx* + *b *to solve
for *y*_{0}. Doing this, we see that

Now, we want to
find the distance between the points (*x*_{0}, *y*_{0})
and (*x*_{1}, *y*_{1}). And so, we use the distance
formula.

And so, the
distance between the point (*x*_{1}, *y*_{1}) and the
line *y* = *mx* + *b* is given by the formula: .

An alternative
form that is often used is ,
where instead of the line *y* = *mx* + *b*, we have the equation
*Ax* + *By* + *C* = 0. If we convert this formula into the
standard slope-intercept form, we have .
And we see that *m *corresponds with and that *b* corresponds with .

Going back to our
formula and replacing *m* and *b*, we have:

.

That is precisely the formula that we wanted to show, and so we are done.