 What we've got now is a two-dimensional proof. We're talking about the Cartesian coordinate system. We have an x-axis and we have a y-axis. Now I've laid out the problem just visually with a couple of points. And this is probably the way that we've expressed this problem since they're not giving you any coordinates. So the problem is this. You have two points on the x and the y-axis, one on each. And this is the information they're giving you. M is the midpoint of AB. And they want us to prove that OM is equal to AM. So the distance from OM is equal to AM. Well, we don't have an M on there, so we better put one on. So M is the midpoint of AB. M is the midpoint of AB. Here's AB and M is going to be the midpoint. So if we draw a line, M is going to be right there. So what do they actually want us to prove? That OM, this distance, is equal to AM. Now they haven't given us any points, so what we're going to do is we're going to make up our own points. So point A here, this point we're going to call 0y. Now keep this in mind. On the y-axis, the x values are always 0. If this is x equals 1, 2, 3, then that's 0, because that goes 0, negative 1, negative 2. So that's the focal point for the x-axis, for the y-axis. No, actually for the x-axis, because that's 0 for the x. And over here, the point for B becomes x and 0. Because just like the y-axis, the y-value on the x-axis is equal to 0. So right there, these are the two points for A and B. So first thing we're going to do is figure out what the midpoint coordinates for the midpoint are. So let's check this out. The midpoint formula, if you remember, is the midpoint of the x-axis and the midpoint of the y-coordination. Now midpoint of the x-axis is just the average of the x-values. So this becomes x1 plus x2 divided by 2. And for the y's, it's just the average of the y's, which is y1 plus y2. Oops, y1 plus y2 over 2. Let's change this up. So we haven't decided which one's going to be our first point, which one's going to be our second. So we're just going to decide it right now. We're going to call this one our first point. This one our second point. So x1 is going to be 0 plus x over 2, because that's our x2. y1 is y. Let's put a comma there. y1 is just y plus y2 is 0 over 2. And the midpoint becomes x over 2 and y over 2. So we can just transfer that information onto the board. x over 2, y over 2. And that's our midpoint. I'm going to erase some of this stuff so we've got more room, but I'm going to keep this information up here. Obviously, when you're doing the problem, you wouldn't be erasing this because this is part of the solution. You're going to get marks for it. x over 2 and y over 2. So they want us to prove that om, the distance from there to there, is equal to am, the distance from there to there. Well, the only way we can do that is use a distance formula. And we've covered that in previous videos. So if you don't remember the formula, you should be going back. You should be memorizing this stuff because these formulas are going to use again and again and again. So the distance formula is square root of x2 minus x1 squared plus y2 minus y1 squared. So let's figure out the distance for om first. And you can actually express it this way. Use subscripts to specify that this is the distance for om. And then later on, we're going to do dam for the distance of am. So the distance of om is going to be... Now, x1 and x2, again, we haven't decided which one of these are x1 and which one is rx2. Now, one other thing we have to remember is the coordinates of the origin is always 00. That's the cross errors where the x-axis crosses the y-axis. So that's going to be 00. So let's decide this to be our first point and this to be our second point. And because we moved on from here, let's just take these guys down because they're no longer our first and second point, right? So x2 is going to be x2, x over 2 plus 0, because that's our x1, squared, plus... Actually, not plus, this is a minus. Don't mess with your formulas. Plus y2, which is y over 2, minus 0 squared. Now, x over 2 minus 0 is just x over 2. So this could come square root of x over 2 squared plus y over 2 squared. And this guy's going to be x squared over... When you do squares, this applies to this and this. So that becomes x squared and 2 squared is 4 plus y squared over 4. And you can just simply write this as square root of x squared plus y squared over 4. Now, you can take this step, take this down one more level to simplify it because this basically becomes the square root of x squared plus y squared over the square root of 4. Now, you can't separate these guys into their square roots because they're added together. And once there's a plus or minus sign between two numbers or two variables, they're married, they're joined at the hips. You can't separate them with the square root. But when it's division like this, it's actually take the square root symbol and take the square root of the top and the square root of the bottom because it's division. So addition or subtraction attaches to things. Division and multiplication, you can separate them. So square root of 4 is just 2. So this just becomes square root of x squared plus y squared over 2. So we just figured out what the distance of O M is. It's x squared, square root of x squared plus y squared over 2. So what I'm going to do, I'm going to erase these guys and put that up here because we know what that answers.