To also find the points,
which trisect the line segment within points
negative two comma negative two,
and six comma five,
which I've already sketched on the coordinate plane.
The two points that trisect the line segment
would be the two points on the line segment where
if they cut the line segment at those two points,
it would create three shorter line segments
all with the same length.
So for example, at one point it might be approximately here,
we'll call it Psub One.
Another point might be approximately here,
we'll call it Psub Two,
where line segment a Psub one
has the same length as line segment
Psub One, Psub Two,
which has the same length as line segment Psub Two B.
To determine the ordered pairs for Psub One and Psub Two,
we will first determine the change of X and change of Y
from point a to point B.
And then because you want to trisect the line segment,
we'll divide those changes by three,
which will give us the change of accent change of Y from
point a to point Psub One,
from point Psub One to point Psub Two,
and from point Psub Two to point B.
So we'll call the point on the left X sub one,
comma Y sub one.
And the point on the right, X sub two comma Y sub two.
The change of X from point a to point B is X sub
two minus X sub one, which is six minus negative two.
Well six minus negative two is equivalent to six plus two,
which is equal to eight.
The X coordinate increases by eight from point a to point B.
So now we'll find Delta Y or the change of Y,
which is equal to Y sub two minus Y sub one,
which is five minus negative two,
which is equal to five plus two, which is seven.
And now we'll find a change of X divided by three.
And the change of Y divided by three.
Well, the change of X divided by three is eight thirds,
eight thirds is equal to two and two thirds
as a mixed number,
eight thirds is equal to six thirds plus two thirds
and six thirds is two.
The change of Y divided
by three is seven thirds.
Which is equal to two and one third.
So now to find point P sub One,
because the change of X divided by three is eight thirds
or two and two thirds.
We add eight thirds or two and two thirds to the X
coordinate of point a,
which gives us negative two plus either eight thirds
or two and two thirds,
I'm going to use eight thirds.
And to find the Y coordinate of point of Psub One,
because the change of Y divided by three is
seven thirds or two and one third,
we add seven thirds or two and one third to the
Y coordinate of point a,
which gives us negative two plus let's use seven thirds.
Let's go and find these sums.
Notice how here the common denominator is three,
as well as here.
Here we have negative six thirds plus eight thirds,
which is two thirds.
And the Y coordinate is negative six thirds
plus seven thirds, which gives us one third.
The ordered pair for P sub One at this point here is
two thirds comma one third.
And then to find point Psub Two
we begin with the ordered pairs for P sub One,
and again add eight thirds or two and two thirds
to the X coordinate.
And seven thirds or two and one-third to the Y coordinate.
So the X coordinate for a P sub Two is going to be
the X coordinate of P sub One plus
change of X divided by three, which is eight thirds.
The Y coordinate is going to be one third,
the Y coordinate of P sub One,
plus the change of Y divided by three,
which is seven thirds.
And now determining the sums for the X coordinate two thirds
plus eight thirds is 10 thirds.
And the Y coordinate is one third plus seven thirds,
which is eight thirds.
If you do want to plot these on the coordinate plane,
it is helpful to have mixed numbers.
10 thirds equal to three and one third
and eight thirds is equal to two and two thirds.
Let's go ahead and check this on the coordinate plane.
The ordered pair for P sub One
is two thirds comma one third.
Which looks about right.
And the order pair for a P sub Two is three and one third
comma two and two thirds,
or for consistency, 10 thirds,
comma eight thirds, which also looks about right.
I hope you found this helpful.