Video: Finding Unknown Coordinates of Points Using the Midpoint Formula

Video Transcript

Find the point 𝐴 on the 𝑥-axis and 𝐵 on the 𝑦-axis such that three over two, negative five over two is the midpoint of 𝐴𝐵.

So in this question, we’re told the midpoint of the line because we’re told that it’s three over two, negative five over two. And this, coupled with a formula for the midpoint of a line, will help us to find the point 𝐴 and the point 𝐵. So the formula for the midpoint of a line is 𝑥 one plus 𝑥 two over two, 𝑦 one plus 𝑦 two over two. And that gives us our 𝑥- and 𝑦-coordinates respectively. And this is where 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two are the coordinates of two different points, so the two endpoints on our line.

In our question, the two endpoints are 𝐴 and 𝐵. So therefore, we’re gonna have 𝑥 one, 𝑦 one for point 𝐴 and 𝑥 two, 𝑦 two for point 𝐵. Before we even need to use our midpoint formula, what we can do is find two of our coordinates just from the information in the question. The first one is one of our 𝐴-coordinates. And that’s because it tells us that the point 𝐴 is on the 𝑥-axis. Well, if we mark any point on the 𝑥-axis, we can see that its 𝑦-coordinate is going to be zero. So therefore, we can say that 𝑦 one is going to be zero and the 𝑦-coordinate of 𝐴 is going to be zero.

And similarly, if we look at point 𝐵, it tells us that point 𝐵 is on the 𝑦-axis. So therefore, if we mark any point on the 𝑦-axis, we can see at this point the 𝑥-coordinate is going to be zero. So therefore, 𝑥 two is zero and the 𝑥-coordinate of point 𝐵 is zero. So great, we’ve got two of our coordinates already. So now, what we need to do is find our missing values, so 𝑥 one and 𝑦 two.

First of all, we’re gonna find 𝑥 one. And we’re gonna to do that using the first part of our formula, the 𝑥-coordinate portion which tells us that 𝑥 one plus 𝑥 two over two will give us our 𝑥-coordinate. Well, we know that the 𝑥-coordinate is three over two. So therefore, we know that 𝑥 one plus zero, and that’s because 𝑥 two is zero, all divided by two is equal to three over two. Well, therefore, we have 𝑥 one over two equals three over two. So we can multiply each side of our equation by two. And when we do this, we get 𝑥 one is equal to three. So therefore, we found the first of our missing coordinates.

Okay, so now we can move on to 𝑦 two. And let’s find that. Well, to find 𝑦 two, what we’re gonna use is the second part of our formula. And that’s 𝑦 one plus 𝑦 two over two. And this is because this tells us our 𝑦-coordinate. And we know that the 𝑦-coordinate of the midpoint is negative five over two. So therefore, we know that zero, and that’s zero because 𝑦 one is equal to zero, plus 𝑦 two over two is equal to negative five over two. So once again, what we do is multiply each side of the equation by two. And when we do that, we get 𝑦 two is equal to negative five. So therefore, we found our other missing coordinates.

So from this, what we can say is that the point 𝐴 on the 𝑥-axis and the point 𝐵 on the 𝑦-axis, such that three over two, negative five over two is the midpoint of 𝐴𝐵, are three, zero and zero, negative five, respectively.