We're asked to find the points
that trisect the line segment within points
negative three comma six
and nine comma negative three.
Notice how I've already sketched the line segment
on the coordinate plane.
The points that trisect a line segment
would be the points on the line segment
where if we cut the line segment at those points,
it would create three shorter line segments
all with the same length.
So for example, for segment AB,
one point might be this point here.
Let's call it P1.
And the second point might be, let's say, here.
We will call it point P2.
Where if we cut the line statement at these two points,
it would create three lane segments
all with the same length,
or we can say the length of line segment AP1
has the same length as line segment P1P2,
which has the same length as line segment P2B.
To determine the ordered pairs
for P1 and P2,
we will begin by determining the change of x
and change of y from point A to point B
and then we'll divide those changes by three,
to determine the change of x
and change of y from point A to point P1,
from point P1 to point P2,
and from point P2 to point B.
Let's label point A x1 comma y1
and point B x2 comma y2.
The change of x is equal to x2 minus x1.
The change of y is equal to y2 minus y1.
X2 minus x1 is nine minus negative three,
which is equal to nine plus three,
which is equal to 12.
X increases by 12 from point A to point B.
Delta y is equal to y2 minus y1,
which gives us negative three minus six,
which is equal to negative nine
Y decreases by nine from point A to point B.
And now we'll divide these changes by three
'cause we're looking for the points
that trisect the line segment.
The change of X divided by three is 12 divided by three,
which is four,
and delta y divided by three
or the change of y divided by three
is negative nine divided by three,
which is equal to negative three.
So now we know the change of x is four
and the change of y is negative three
from A to P1, from P1 to P2,
and from P2 to B.
To show this, starting at point A,
if the change of x is four,
we would go right four units.
Because the change of y is negative three,
we would go down three units
to find the ordered pair for P1.
From P1 to P2, we would do the same.
We would go right four units
because the change of x is positive four,
down three units because the change of y is negative three,
and just to verify if we go right four
and down three again,
we are at the endpoint, point B.
And now let's calculate the ordered pairs for P1 and P2.
To find the ordered pair for P1,
we start at point A,
so x is negative three,
and then add delta x divided by three,
which is positive four.
The y coordinate starts at six
and then decreases by three,
so we add negative three,
which gives us P1 has an x coordinate
of negative three plus four, positive one,
and a y coordinate of six plus negative three,
which is positive three.
And now to find the ordered pair for P2,
we had delta x divided by three to the x coordinate of P1,
and we add delta y divided by three, or negative three,
to the y coordinate of P1.
So P2 has an x coordinate of one plus four
and a y coordinate of three plus negative three.
Simplifying, the x coordinate is five
and the y coordinate is zero.
And our segment does verify this result.
Notice how P1 does have an ordered pair of one comma three
and P2 does have an ordered pair of five comma zero.
I hope you found this helpful.