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# Dividing line segments

Watch Sal figure out the coordinates of a point between two other points that give a certain ratio. Created by Sal Khan.

Video transcript

A, B and C are
collinear, and B is between A and C. The ratio
of AB to AC is 2 to 5. If is A is at negative 6 comma
9, and B is at negative 2 comma 3, what are the
coordinates of point C? And I encourage you to
now pause this video and try this on your own. So let's try to visualize this. So A is at negative 6 comma 9. B, let's see, it's less negative
in the horizontal direction, it's lower in the
vertical direction. So we could put B
right over here. B is at the point
negative 2 comma 3. And that C, it's
going to be collinear, so we're going to go
along the same line, let me draw that line right now. So it's going to be
collinear and they tell us that the ratio of
AB to AC is 2 to 5. So B is going to
be 2/5 of the way. So let's say C is--
I'm just trying to eyeball it right now--
let's put C right over here. And we don't know
C's coordinates. Well the way we
could think about it is to break it up into
horizontal change in coordinate and vertical change
in coordinate, and apply the same ratio. So for example, what is
the horizontal change in coordinates
going from A to B? Well let's draw that. Going from A to B, so
this is A's x-coordinate, it's at negative 6. B's x-coordinate
is at negative 2. So it's this change
right over here. This is the horizontal
change that we care about. Now what is that? Well if you start at negative
6 and you go to negative 2, you have increased by 4. Another way of thinking about it
is negative 2 minus negative 6 is the same thing as negative 2
plus 6, which is going to be 4. Now the ratio between
this change and the change of the x-coordinate between A
and C is going to be 2 to 5. So let's call that change,
this entire change, let's call this x. So we could say that the ratio
between 4 and x is equal to-- and notice this is the
horizontal change from A to B just if you look on
the horizontal axis-- so the ratio of
that, which is 4, to the horizontal
change between A and C, well that's going to have
to be the same ratio. So it's going to be 2/5. Now to solve for x,
a fun thing might be to just take the
reciprocal of both sides. So x/4 is equal to 5/2. We can multiply
both sides times 4 and we are left with x is equal
to 5 times 4 divided by 2, which is equal to 10. So the change in x from
A to C is going to be 10. So what does that tell us
about C's x-coordinate? So we could start with
A's x-coordinate, which is negative 6, add 10 to
it, negative 6 plus 10 is 4. So we figured out
the x-coordinate, now we just have to do
the same thing for the y. So what is the change
in y going from A to B? Well here, we go from 9
to 3, we've gone down 6. Another way you could say it
is, 3 minus 9 is negative 6. To find the change,
you could think, I'm just taking the end
point and subtracting from that the starting point. Negative 2 minus negative
6 was positive 4. 3 minus 9 is negative 6. Or you could just look at it. We've gone down 6, so we
can write negative 6 here. Now our change in y,
we're going to have to have that same ratio. So our change in
y between A and C, let's just call that distance y. So our change in y is,
that's our change in y. And we're going to have
to have the same ratios. So we could write that the
ratio between our change in y from A to B, which
is negative 6, to the ratio between
our change in y from A to C, negative
6 to y, is once again going to be equal to 2 to 5. Once again, we could take
the reciprocal of both sides, y over negative 6 is
equal to 5 over 2, multiply both sides
times negative 6, and we are left with y is
equal to 5 times negative 6 is negative 30, divided
by 2 is negative 15. So our change in y,
or I guess our change in our vertical axis, which
we're calling y in this case, is negative 15. So here, if our y
value is 9 and if we were to subtract 15 from
that, where does that put us? Well 9 minus 15 is
going to put us at 6. So the coordinates for point C
are at-- oh, sorry 9 minus 15 is going put us at negative 6. I almost made a
careless mistake. 9 minus 15 is negative 6. I was wondering, this seems
very low to be at 0.6. So 4 comma negative 6 is
the coordinates of point C.