Question Video: Solving a Word Problem by Dividing a Line Segment Mathematics

A bus is traveling from city A (10, −10) to city B (−8, 8). Its first stop is at C, which is halfway between the cities. Its second stop is at D which is two-thirds of the way from A to B. What are the coordinates of C and D?

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Video Transcript

A bus is traveling from city A 10, negative 10 to city B negative eight, eight. Its first stop is at C, which is halfway between the cities. Its second stop is at D, which is two-thirds of the way from A to B. What are the coordinates of C and D?

Well, the first thing we’ll have a look at is stop C cause we can see that this is halfway between the cities. Well, if we’re looking to find the midpoint between any two points, we have a formula to help us. And that formula is that the coordinates of a midpoint is equal to, then we have 𝑥 sub one plus 𝑥 sub two over two. So that is the 𝑥-coordinates of both of our points added together then divided by two. And then for the 𝑦-coordinate, we have 𝑦 sub one plus 𝑦 sub two over two, which is the 𝑦-coordinates added together and then divided by two.

Well, now that we have the formula, what we can do is use it to help us find the midpoint of our bus travel from city A to city B, which is stop C. And so that we can achieve that, what we need to do is label our coordinates, which we’ve done here. So we’ve got 𝑥 sub one, 𝑦 sub one; 𝑥 sub two, 𝑦 sub two. So now, let’s substitute these into the formula. And when we do that, we’re gonna have that C, our midpoint, is going to be, then we’ve got for the 𝑥-coordinate 10 add negative eight because that’s 𝑥 sub one add 𝑥 sub two. And then this is divided by two. And then for the 𝑦-coordinate, we have negative 10 plus eight divided by two.

So when we calculate this, what we’re going to get is two over two or two divided by two for our 𝑥-coordinate and negative two over two for our 𝑦-coordinate. And it’s worth noting that we got two over two or two divided by two for our 𝑥-coordinate because we had 10 and negative eight. And if you add a negative, it’s the same as subtracting a positive. So it’s the same as 10 minus eight, which will give us our two. So when we do the calculation, we’re gonna get one, negative one. So this is going to be the coordinates of C, which is the halfway point between city A and city B.

So now, what we’re gonna have a look at is point D. And we know that stop D or point D is two-thirds of the way from A to B. Now, one way that we could work this out to find out the position of D would be to find out what two-thirds of the distance of the 𝑥-coordinate from A to B is and then two-thirds of the distance of the 𝑦-coordinate from A to B. And then add this distance onto the original 𝑥- and 𝑦-coordinate of A. However, there is a formal way that we could do that using a formula.

And the formula tells us that if we want to find the point 𝑥, 𝑦, which is between two points, then this is equal to 𝑥 sub one plus 𝑘 multiplied by 𝑥 sub two minus 𝑥 sub one. Then for the 𝑦-coordinate, 𝑦 sub one plus 𝑘 multiplied by 𝑦 sub two minus 𝑦 sub one, where 𝑘 is the fraction of the total length or distance between the two points. Okay, great. So we have this formula. Let’s use it to help us find out the position of point D.

So then we can say that point D is at the point where we’ve got the 𝑥-coordinate with 10 plus two-thirds multiplied by negative eight minus 10, where two-thirds is our 𝑘. Then for the 𝑦-coordinate, we’ve got negative 10 plus two-thirds multiplied by eight minus negative 10. And two-thirds is our 𝑘 cause we’re told that D is two-thirds of the way between A and B. So what we get for this, for the 𝑥-coordinate, we’ve got 10 plus two-thirds of negative 18. Then for the 𝑦-coordinate, we’ve got negative 10 plus two-thirds of 18. So this is gonna give us 10 minus 12 for the 𝑥-coordinate and negative 10 plus 12 for the 𝑦-coordinate.

So therefore, the final coordinates for our point D are going to be negative two, two. So therefore, we can say the coordinates of points C and D are one, negative one and negative two, two, respectively.