Welcome to a lesson on determining the perpendicular bisector
of a segment on the cordinate plane. Here we see a red perpendicular bisector
of the blue segment on the coordinate plane. Remember that tells us two
things,
first this red line cuts the segment into two congruent segments,
or two segments that have the same length. So this segment here
is the same length as this segment here. And the second thing
is that the intersection of the line in this segment form a right angle.
So this angle right here is a right angle, and therefore measures ninety degrees.
And in fact, all four of these angles would be right angles.
So the goal of this video is to determine the equation
of this red perpendicular bisector given
a segment on the coordinate plane. In order to do this, there are two main things we have
to do.
Number one, if we determine the midpoint
of the segment, notice that'll give us one point on the perpendicular bisector.
And the second thing to remember is that, if two lines are perpendicular,
their slopes are negative reciprocles. Notice how the slope of this blue segment
is
positive one, and the slope of the perpendicular bisector
is negative one. They're negative reciprocals of one another
and their product is also equal to negative one. Let's go and take a look at an example.
So here we want to determine the equation of the perpendicular bisector
of the given segment. Let's start by determining the coordinates of the end
points of the given segment.
This point here would be negative three, negative four.
This point here would be five, zero. So the first step is going to be to determine the
midpoint of the given segment. Remember that's going to give us one point on the
perpendicular bisector.
So to determine the x-coordinate of the midpoint,
we sum the x-coordinates, and then divide by two.
So we'll have negative three plus five, divided by two.
Then to determine the y-coordinate of the midpoint, we add the y-coordinates and
divide by two.
So the x-coordinate of the midpoint is going to be negative three plus five, that's two divided by two,
that's positive one.
And here we'll have negative four divided by two, that's negative two.
The next thing we have to do is determine the slope
of the segment, because the slope of the line is going to be the negative reciprocal.
So the slope of this segment, we can either use the formula:
y--two minus y-one divided by x-two
minus x-one. Or we could try to pick two convenient points
on the coordinate plane. I'm going to go ahead and review the formula.
It's y-two divided by y-one, divided by x-two
minus x-one.
So let's call these the ones and these the twos.
So we'll have zero minus negative four,
over five
minus negative three.
So we'll have positive four
over positive eight, so the slope of the segment
is equal to positive one-half. So the perpendicular bisector of this segment
is going to pass through the point
one, negative two.
And the slope of the perpendicular bisector
is going to be the negative reciprocal of one-half. So if we flip this over and change
the sign, we'll have negative two over one,
or just negative two. So now we'll determine the equation of this line.
Remember to find the equation of a line given
the slope and a point, we can either use the sloped intercept form, or we can use the
point slope form of the line.
Let's go ahead and use the point slope form of a the line.
So if we do this, our slope m is negative two,
x-one is positive one, and y-one is equal to negative two.
So we'll have y minus
y-one that's minus negative two, must equal the slope
negative two, times x minus
x-one. So now we'll go ahead and solve this for y to put it in slope intercept form.
So we'll have y plus two equals negative two-x,
plus two. And now if we subtract two on both sides,
we have y equals negative two-x.
So we have a slope of negative two, and our y-intercept would be zero.
Let's go ahead and graph that to see if it is the perpendicular bisector.
Well we have a y-intercept of zero, and a slope of negative two.
To graph it we'd probably put negative two over one, so if we go down two and right one,
notice we're at the midpoint one, negative two
of the segment. So this does bisect it
and that line does form a right angle
with the blue segment. So that is the equation of the perpendicular bisector:
y equals negative two-x.
I hope you found this example helpful.