Question Video: Finding the Distance between given Points Using the Distance between Two Points Formula Mathematics • 6th Grade

Find the lengths of 𝐴𝐵 and 𝐷𝐶, where the coordinates of points 𝐴, 𝐵, 𝐶, and 𝐷 are (−2, 3), (5, 3), (−2, −4), and (−2, −5), respectively, considering that a length unit is equal to 1 cm.

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Video Transcript

Find the lengths of the line segment 𝐴𝐵 and the line segment 𝐷𝐶, where the coordinates of points 𝐴, 𝐵, 𝐶, and 𝐷 are negative two, three; five, three; negative two, negative four; and negative two, negative five, respectively, considering that a length unit is equal to one centimeter.

So the first thing we’ve done is marked our line segments onto our diagram, so we’ve got 𝐴𝐵 and 𝐷𝐶. And as we actually have a horizontal line for 𝐴𝐵 and a vertical line for 𝐷𝐶, then what we can do is use a couple of methods to solve the problem. The first method is the most straightforward method, but I also want to show you a more formalized method just to show you how a formula could be used for this type of problem as well. Well, in order to find the length of the line segment 𝐴𝐵, because it’s horizontal, what we need to do is look for the change in the 𝑥-coordinates.

So therefore, we can find this by having the 𝑥-coordinate of 𝐵 and subtracting from it the 𝑥-coordinate of 𝐴. So we have five minus negative two, which will give us an answer of seven centimeters. And we can check this by counting on the squares on our diagram. So we got here seven, so it would be seven centimeters.

Okay, great! So now let’s have a look at the line segment 𝐷𝐶. Well, this time as we’re looking at the length of the line segment 𝐷𝐶, which is a vertical line, we’re gonna be looking at the change in our 𝑦-coordinates. So what we’re gonna have is negative four minus negative five, which will give us the answer one. So we know that it’s one centimeter long. And again, we could check that out by counting the squares that we have on our axis, knowing that every two squares is equal to one unit or one centimeter.

It is worth addressing at this point what would happen if you took the 𝑦-coordinates the other way around, so we had negative five minus negative four. Well, this would give a result of negative one. And what we can do is disregard the negative, and that’s because we’re only interested in the magnitude because we’re looking at distance. So therefore, we would just get one, which would be one centimeter as well. Now, the reason we have it the way around that we have in the question and in our answer is because to calculate how far it is from 𝐷 to 𝐶, we usually start with the coordinates of 𝐶 and then subtract the coordinates of 𝐷.

So, we’ve solved the problem as we said; however, what we’re also gonna look at is a formalized way of doing it using the distance formula. And what the distance between two point formula tells us is that the distance between 𝐴 and 𝐵 is equal to the square root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴 all squared plus 𝑦 sub 𝐵 minus 𝑦 sub 𝐴 all squared. And this is from the Pythagorean theorem.

Well, to demonstrate how this would work for our 𝐴𝐵, so our line segment 𝐴𝐵, this would be equal to the square root of five minus negative two all squared, so that’s the change in our 𝑥-coordinates, plus three minus three all squared, which would just give us the square root of seven squared. And that’s because for the second part, we’ve got three minus three, which is just zero, and zero squared is just zero, which would just be the square root of 49. So it’d just be seven. Also, we know if we have the square root of a squared number, then it would just be the number itself. So great! That’s 𝐴𝐵.

So now let’s have a look at 𝐷𝐶 using this method. So then for 𝐷𝐶, what we’re gonna have is square root of negative two minus negative two all squared plus negative four minus negative five all squared, which would just give us root one squared. So this would just give us root one, which just gives us the answer of one. So we’ve got the same answer as before. It is worth noting that we’ve got root 49 and root one. It would usually have two results, positive or negative, so positive or negative seven or positive or negative one. However, in this instance, we’re just looking at the positive result because we’re looking for a length or a magnitude.

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