Video Transcript
Suppose 𝐴 one, three and another point 𝐵 and that 𝐶 five, one divides
the line segment 𝐴𝐵 internally in the ratio two to three. What are the coordinates of 𝐵?
So what we’ve done first of all is
drawing a little sketch, so we’ve got our point 𝐴 and our point 𝐶. And both of these lie on the line
segment 𝐴 to 𝐵, but we don’t know where 𝐵 is. But we do know that the ratio two
to three is what divides our line segment 𝐴𝐵. So if we consider our line segment
𝐴, 𝐶, and 𝐵, we know that 𝐶 lies between 𝐴 and 𝐵 because we’re told that 𝐶
divides the line segment 𝐴𝐵 internally. And we know that the ratio is two
to three, so we know the distance between 𝐴𝐶 represents two parts of our ratio and
the distance between 𝐶 and 𝐵 represents three parts.
So therefore, what we’re gonna do
is split this up and take a look at the 𝑥-coordinates and then the
𝑦-coordinates. So we’re gonna start with the
𝑥-coordinates. So what we want to do is we want to
find the difference between the 𝑥-coordinates of 𝐴 and 𝐶. So I’ve labeled 𝐴 𝑥 sub one, 𝑦
sub one and 𝐶 𝑥 sub two, 𝑦 sub two. So we’re gonna start with finding
the difference between our 𝑥-coordinates. So we’ve got 𝑥 sub two minus 𝑥
sub one, which is gonna be equal to five minus one, which is gonna be equal to
four. So we know that the difference
between the 𝑥-coordinates of 𝐴 and 𝐶 is four.
So if we take a look at our ratio,
we can see that it’s two to three. So we know that two parts is equal
to four. So if we divide this by two, we can
find that one part is gonna be equal to two. So we know that one part of the
ratio is equal to two. Well, if you want to find the
difference between the 𝑥-coordinates of 𝐴 and 𝐵, we want the whole of our ratio,
so we want both parts. So we’ve got our two and our three,
which is a total of five parts. So therefore, we want to find out
what five parts is gonna be if one part is equal to two. So therefore, if we multiply both
sides by five, we get that five parts is equal to 10. So therefore, we know that the
difference between the 𝑥-coordinates of 𝐴 and 𝐵 are gonna be 10.
So therefore, if we called the
coordinates of 𝐵 𝑥, 𝑦, we could find the 𝑥-coordinate by adding the
𝑥-coordinate of 𝐴, which is one, to the difference, which is 10, which is gonna
give us 11. Okay great. So now we can move on to the
𝑦-coordinates. Well, the difference between our
𝑦-coordinates is gonna be 𝑦 sub two minus 𝑦 sub one, which is equal to one minus
three, which is equal to negative two.
So once again, we know that two
parts is equal to negative two. So then what we’re gonna do is
divide this by two to find out what one part is. That’s gonna be one part is equal
to negative one. So therefore, if we multiply this
by five, we’re gonna get that five parts is equal to negative five. So therefore, what that means is
the difference between the 𝑦-coordinates of 𝐴 and 𝐵 is going to be negative
five. So therefore, to find the
𝑦-coordinate of 𝐵, what we’re gonna do is take the 𝑦-coordinate of 𝐴 and
subtract five from it, which will give us negative two.
So therefore, we can say that the
coordinates of point 𝐵 are 11, negative two.
