Given the point 𝐴 negative five, nine and the point 𝐵 seven, negative three, what are the points 𝐶 and 𝐷 that divide the line segment 𝐴𝐵 into three parts of equal length?
So the first thing we’ve done is drawn a sketch for our line segment 𝐴𝐵. And we’ve done that by plotting the point 𝐴 negative five, nine and the point 𝐵 seven, negative three. So what we’ve done here is just model the situation using a horizontal line. So we’ve got the line segment 𝐴𝐵. Then we’ve got two points within it that split it up into three equal lengths. So we’ve got a third of the length, a third of the length, and a third of the length. So therefore, from this diagram, we can see that the point 𝐶 would be a third of the way along the line. And the point of 𝐷 would be two-thirds of the way along the line.
Now, we have two different ways that we can solve this problem. The first way we’re gonna use is by having a look at the coordinate grid. So first of all, if we count how many units it is down from 𝐴 to 𝐵, so in the 𝑦-direction, we can see that it’s 12. And if we do the same in the 𝑥-direction, so we see how many units along from 𝐴 to 𝐵 is, it is also 12. Well, if we start by looking at point 𝐶, well, we know that point 𝐶 is a third of the distance along the line from 𝐴 to 𝐵. So therefore, we can find a third of the way along from 𝐴 to 𝐵, which will be four squares down because a third of 12 is four. And it’ll be the same along the 𝑥-axis because it’ll be four units along because it’s a third of 12, which, once again, is four.
So now what we can do is go four down from 𝐴 and four along from 𝐴. And then what we do is find out where that meets. Well, that point where they meet is the point 𝐶, which is at the coordinates negative one, five. Well now if we look at the point 𝐷, what we’re looking at is a point that is two-thirds along the way from 𝐴 to 𝐵. So, I’ll mark this again on the diagram, so we got two-thirds along. So that’s eight units down and eight units cross. So then, these meet at the point which we’re gonna call 𝐷, which is the point three, one. So now great! What we’ve done is we’ve found the two points 𝐶 and 𝐷 that divide the line segment 𝐴𝐵 into three parts of equal length.
So we did say that there were a couple of methods we could use. So we’re gonna take a look at another method. And in this method, we’ll actually formalize the process using a formula. So, the formula we have is that if we want to find the point 𝑥, 𝑦 along a line segment, then this is equal to 𝑥 sub one plus 𝑘 multiplied by 𝑥 sub two minus 𝑥 sub one. And that’s for the 𝑥-coordinate. And for 𝑦-coordinate, we have 𝑦 sub one plus 𝑘 multiplied by 𝑦 sub two minus 𝑦 sub one. And this is where 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two are the coordinates of the two points of the line segment and, in this case, that would be 𝐴 and 𝐵. And 𝑘 is the fraction along or fraction of the distance or length of that line segment to the point you’re looking for is.
Okay, so now let’s substitute in our values into our formula. So the first thing we’ve done is label the coordinates we’ve got. So we’ve got 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. So now we’re gonna have a look at the point 𝐶, which is a third of the way along our line. So when we do that, for the 𝑥-coordinate, we’re gonna have negative five plus a third multiplied by seven minus negative five. Then for the 𝑦-coordinate, we’re gonna have nine plus a third multiplied by negative three minus nine.
And as we said before, this is just formalizing what we did with the previous method because in the parentheses on the left-hand side, we’ve just got the change in the 𝑥-coordinates. So that means we want a third of the way along that change. And on the right-hand side in the parentheses, we have the change in the 𝑦-coordinates, so we want a third of the way along that change. So, when we calculate these values, we’re gonna have negative five plus four, nine minus four, which will give us the coordinates negative one, five, which match what we got with the first method.
So now quickly we’re going to the point 𝐷. Well, for the point 𝐷, everything is gonna be exactly the same, except this time we’ve got two-thirds is our 𝑘 because it’s two-thirds of the way along the line segment 𝐴𝐵. So we got negative five plus two-thirds multiplied by seven minus negative five for the 𝑥-coordinate and nine plus two-thirds multiplied by negative three minus nine for the 𝑦-coordinate which, when calculated, will give us the coordinates three, one, which is the same as we got with the first method.
So therefore, we found the points 𝐶 and 𝐷 that divide the line segment 𝐴𝐵 into three equal parts, and they are negative one, five and three, one, respectively.