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## Dividing line segments

# Dividing line segments: graphical

CCSS.Math:

## Video transcript

Find the point B on segment AC,
such that the ratio of AB to BC is 3 to 1. And I encourage you
to pause this video and try this on your own. So let's think about
what they're asking. So if that's point
C-- I'm just going to redraw this line segment
just to conceptualize what they're asking for. And that's point A. They're
asking us to find some point B that the distance
between C and B, so that's this distance
right over here. So if this distance is x, then
the distance between B and A is going to be 3 times that. So this will be 3x. That the ratio of
AB to BC is 3 to 1. So that would be the ratio--
let me write this down. It would be AB-- that looks like
an HB-- it would be AB to BC is going to be equal
to 3x to x, which is the same thing as 3 to
1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted
to say, oh, well, you could use the
distance formula to find the distance,
which by itself isn't completely uncomplicated. And then this will
be 1/4 of the way. Because if you think about
it, this entire distance is going to be 4x. Let me draw that a
little bit neater. This entire distance, if
you have an x plus a 3x, is going to be 4x. So you'd say, well, this is 1
out of the 4 x's along the way. This is going to be 1/4 of
the distance between the two points. Let me write that down. This is 1/4 of the
way between C and B, going from C to A. B is
going to be 1/4 of the way. So maybe you try to
find the distance. And you say, well, what
are all the points that are 1/4 of the way? But it has to be 1/4
of that distance away. But then it has to
be on that line. But that makes it
complicated, because this line is at an incline. It's not just horizontal. It's not just vertical. What we can do, however,
is break this problem down into the vertical
change between A and C, and the horizontal
change between A and C. So for example, the horizontal
change between A and C, A is at 9 right over here,
and C is at negative 7. So this distance right over
here is 9 minus negative 7, which is equal to 9 plus
7, which is equal to 16. And you see that here. 9 plus 7, this total
distance is 16. That's the horizontal distance
change going from A to C, or going from C to A.
And the vertical change, and you could even just count
that, that's going to be 4. C is at 1. A is at 5. Going from 1 to 5, you've
changed vertically 4. So what we can say, going
from C to B in each direction, in the vertical direction
and the horizontal direction, we're going to go
1/4 of the way. So if we go 1/4 in the
vertical direction, we're going to end up
at y is equal to 2. So I'm just going, starting
at C, 1/4 of the way. 1/4 of 4 is 1. So I've just moved up 1. So our y is going
to be equal to 2. And if we go 1/4 in the
horizontal direction, 1/4 of 16 is 4. So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that
point right over there. We end up at this point. This is the point
negative 3 comma 2. And if you were really
careful with your drawing, you could have actually
just drawn-- well, actually you don't have
to be that careful, since this is graph paper. You actually could
have just said, hey, we're going to go 1/4 this way. Where does that
intersect the line? Hey, it intersects the
line right over there. Or you could have said, we're
going to go 1/4 this way. Where does that
intersect the line? And that would have let you
figure it out either way. So this point right
over here is B. It is 1/4 of the
way between C and A. Or another way of thinking about
the distance between C and B, which we haven't
even figured out. We could do that using
the distance formula or the Pythagorean theorem,
which it really is. This distance, the distance
CB, is 1/3 the distance BA. The ratio of AB to BC is 3 to 1.