Given 𝐴 four, eight and 𝐵 six, six, what are the coordinates of the midpoint of 𝐴𝐵?
To enable us to understand what’s happening here, I’ve drawn a little sketch. So I’ve drawn our coordinates. So we’ve got 𝐴 four, eight and 𝐵 six, six. And I’m drawing the line between them. And that’s what the straight line above the 𝐴𝐵 means. It means the line 𝐴𝐵. And what we’re actually trying to find is the midpoint, so I’ve marked that on here with an 𝑚, of that line.
And to enable us to do that, we actually have a formula that’s gonna help us. And this formula tells us that the midpoint 𝑚 is equal to 𝑥 two minus 𝑥 one over two, to give us our 𝑥-coordinate, and 𝑦 two minus 𝑦 one over two, to give us our 𝑦-coordinate.
Okay, great! So we know we can use that because we actually have a pair of coordinates, 𝐴 and 𝐵. To help us further, what I’ve actually done is that I’ve actually labelled our coordinates so that we can see now what 𝑥 one, 𝑦 one, 𝑥 two, and 𝑦 two actually are so that we are now able to substitute them back into our formula.
So that’s gonna be our next step. So we get 𝑚 is equal to and then as the numerator we have six, because that’s 𝑥 two, minus four, because that’s 𝑥 one, and that’s all gonna be divided by two. And then we’ve got six, because that’s 𝑦 two, minus eight, because that’s 𝑦 one.
And again this is all divided by two, which then in turn gives us two over two for our 𝑥-coordinate and minus two over two for our 𝑦-coordinate. And now we can get the answer to these to find out our midpoint.
So therefore, we can say that our midpoint is equal to one minus one. So our 𝑥-coordinate is one and our 𝑦-coordinate is minus one. And I’ve also shown it again on our graph. So that’s the problem solved.
So quick recap! First thing is make sure if you’ve got two points, we know that we can actually use the midpoint formula, then label your points from your coordinates so you know the 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two.
And then we substitute that back into the formula. And it’ll give us our two coordinates, 𝑥 and 𝑦, for our midpoint.