 Section 10.8 is all about equations of circles. And so we're going to do a little exploring here to find the standard equation of a circle. Just like a straight line has a standard equation, which can be y equals mx plus b, a circle has a standard equation. And we're going to see how this standard equation of a circle is derived. If we look at this first graph here, you can see pretty clearly that if I were to connect these dots, we'd make a circle. But we want to do a little exploring here to help us find the equation. So the first four problems, we're just asked to find the distances of four segments on our graph. gc is going to be this distance, this segment right here. And if we count, because it's a horizontal line, the distance is just one, two, three, four, five units long. The second one, lc, is now going to be this segment on the opposite side. And again, because it's a horizontal line, we can just count. And when you do that, you're going to get five for lc as well. And you can probably guess number three and four, nc. We're going to now do the vertical line from n to c. When you count that, that's going to be five units. And jc, we're going to do the same thing. This j value up here is off-center a little bit, it looks like. But when we count that, that will be five as well. And so because we're just working with horizontal and vertical segments, we can just count the boxes on our graph. When we go to the next number, five and number six, however, we see that hc is that diagonal segment right there. And so we can't count like we did on the first four. This is where we're going to use, yes, the distance formula. We're going to find the distance between h and c. And remember, if it's helpful, because these are really easy to get mixed up on, if it's helpful for you, you can put in the first xy value, the second xy value, and that can make sure that we don't make little mistakes on these. So when we use the distance formula, we're going to take the x values and find the difference, x2 minus x1. And be careful of that negative negative that we get there. We're going to add the difference of the y values. And don't forget, we're going to take the square root of the whole thing. 1 minus negative 3 is 1 plus 3, which is 4 squared. 3 minus 6 is negative 3. And we're going to square that as well and take the square root of the whole thing. We're going to get 16. Negative 3 squared is a positive 9. And 16 plus 9, 25. Square root of 25, of course, is 5. And we did all that work to show that, yes, this segment is the same distance as the first 4, 5. And we're going to be asked to do that for problem number 6, the distance from k to c. I'm going to do the same thing. I'm going to find the distance from point k to the distance to point c. And when I do that, I'm going to subtract the x values, y minus 1 minus 4 squared. And then find the difference of the y values, 3 minus 7 squared. Don't forget the square root. We're going to get negative 3 squared plus negative 4 squared. And remember, when you're squaring those negative numbers, they're always going to be positive. Negative 3 times negative 3 is positive 9. Negative 4 times negative 4 is positive 16. And so not a big surprise that, again, we're going to get a distance of 5 using the distance formula. And in fact, if you found the distance between point c and any of these points, we won't make you do that with the distance formula, but if we did do that, you would find the distance between any point on that circle and point c will be the same distance. And the next question here, what geometric figure is created by all of the points except for c? Just about that in the beginning, we can visually see that, yes, that is a circle. And we just prove that that distance from the point c to any point on that circle is the same. So if we plotted a point, any point on this geometric figure and found the distance between that point and point c, the distance would be 5, which actually represents the radius of our circle. And that's going to just bring us to deriving the actual standard equation of a circle and the equation of this circle based on what we just did. And so this last question just says, set up the distance formula for a point x, y, any point x, y and point c and set it equal to that distance of 5. And that's going to help us make our standard equation of a circle. And so if we're using the distance formula, we're going to, I'm going to just call this x1, y1 and I'm going to call this x2, y2. And so when I do the distance formula, I'm going to do my x value minus 1 square that and then the same thing, the y minus the 3 squared. And we are going to take the square root of that and when we did that, we found that that was a distance of 5. If we wanted to clean up this equation in order to get rid of a square root of an equation, remember we can just square the whole thing and if we do that on one side of the equation, we're going to do that on the other side. And when we square that, that's going to cancel out this radical and leave behind just what's inside that. x minus 1 squared plus y minus 3 squared equals and when I simplify 5 squared, I'm going to get 25. This is the equation of that circle that we just graphed up above. And we're going to talk about how we derive that from the distance formula and what that means to us as we go along and do more practice problems.