Welcome to a lesson on determining the distance between two parallel lines on
the coordinate plane.
The shortest distance between two parallel lines
is the length of a perpendicular segment, as we see here in red.
If the segment isn't perpendicular to both parallel lines
then they would not be the shortest distance. And notice how there are
several options for
a perpendicular segment. In fact there's an infinite number of perpendicular
segments between
two parallel lines. But we want to determine a perpendicular segment where both endpoints
would have integer coordinates. And we also need to remember that
these red perpendicular segments would be on perpendicular lines.
And remember the slopes of perpendicular lines are negative reciprocals of one another.
Just review that. If we had two perpendicular lines,
let's say the first line had slope negative two-thirds,
the second line is perpendicular to this. The slope would have to be the negative
reciprocal
of negative two-thirds, which means we're flip it over and change the sign.
So the slope of the second line will be positive three-halves.
Looking at another example. If line three and four are perpendicular,
and line three had a slope of five, it's going to be helpful to make five into a fraction,
so we'll put five over one. And if line four is perpendicular to this line,
the negative reciprocal of five over one would be negative
one over five. Now the only exception to this rule is if we have horizontal and
vertical lines. A horizontal line has a slope of zero,
and a vertical line has an undefined slope.
And of course horizontal and vertical lines are always perpendicular.
Let's go ahead take a look at our example.
Here we want to determine the shortest distance between these two parallel lines.
So step one, we need to find a convenient point on either line.
So for example this point here is convenient,
because the coordinates are two, negative three.
Our next step is going to be to determine the slope
of these blue lines. Again, there's a couple ways of doing this. We could use the formula:
m equals y-two-minus y-one
over x-two minus x-one, or we could again just pick two convenient points
on either line, and draw a triangle to determine
the change in y with respect to the change of x, or
rise over run. So if I selected this point here,
and let's say this point here,
I could go ahead and sketch a triangle,
which would show the rise over the run,
or the change in y over the change of x. So notice that I would go up
two units, that would be two for the change in y,
and then I go to the right four units, so the change of x is positive four.
So the slope of this blue line
would be two
over four, or one-half.
The reason this is so helpful is, remember that the
shortest distance between these two parallel lines would be a perpendicular segment.
So the segment would have a slope that would be the negative reciprocal
of one-half. So that would be negative two over one, or negative two.
But negative two over one is going to be actually more helpful.
So what we'll do now is sketch a line
with the slope of negative two that passes through
this red point. And then from that line we'll determine the
endpoints of the perpendicular segment. So using the slope, negative two over one, from
this point,
we'll go down two and right one. Now remember it doesn't matter whether this negative
is in the numerator, or the denominator,
this is the same as to over negative one. Now the reason I mention this is I could go up
two units and left one unit instead of going down two and right one.
Let's do that again. Let's go up two,
and left one, and notice that we just found the point of intersection,
which would be the end point of the perpendicular bisector.
But let's go ahead and sketch the line anyway. So here's our perpendicular segment,
and now we can see that the other end point
is going to have the coordinates zero, one.
So we want to determine the distance between zero, one
and two, negative three. So again, we'll call these the ones,
these the twos,
and now we'll use the distance formula to determine the distance between
the two parallel lines.
So we're going to have
the square root of two minus zero squared,
plus negative three minus one squared.
So we'll have two squared, that's four. Plus this is going to be negative four squared, that's sixteen.
The distance is equal to the square root of twenty.
Remember the square root of twenty can be simplified.
Twenty is two times two, times five,
so we have a perfect square factor here. So the distance is equal to
two square root five, which we approximately have four point
four, seven, two. Now I do want to mention this point of intersection right here was very
convenient because the coordinates were integers.
If it wasn't that easy to determine we'd have to find the equation of this
red line,
and then determine the equation of this blue line here,
and then solve it of the system of equations to determine this point
of intersection.
I hope you found this helpful. Thank you for watching.