 This video is going to talk about circles and all the things that we know about a circle. The first two things that we know about a circle, we want to talk about the midpoint and the distance formula. So given two points, we should be able to find a point in between them, exactly in the middle, and we should be able to find the distance between those two points. And that's this first question. Find the distance. So it's the square root of, and we're going to take the difference of the x's, and it doesn't matter what order you subtract them in because we're going to square it. Mine won't matter. So we take negative 5 minus 3, and we're going to add to that the difference of the y's. So I'm going to take negative 2 minus 4, just so my signs stay the same, and that's squared. So negative 5 minus 3 would be negative 8, that needs to be squared, plus negative 6, that needs to be squared, that's a negative 6. So we have 64 plus 36 that are being added under the radical, which is the square root of 100 or 10. So the diameter of that circle would be 10. Now we want to find the midpoint. Negative 5, 4, 3, negative 2, so I don't forget, that's a 2, and we want to find the midpoint. Well in the midpoint you have to remember that now you're going to add the x's and the y's to find the ordered pair. So for the x's we take negative 5 plus 3, and then you always divide that by 2, that's the x, and then the y's are going to be 4 minus 2, since it's plus a negative 2, and divide that by 2, maybe I'll put plus a negative just to remind you. And then we have here negative 5 plus 3 would be negative 2 over 2, and here 4 minus 2 or plus a negative 2 would be 2 over 2, and simplifying we find out that the midpoint is at negative 1, 1. So then it asks us finally what the radius of the circle is. Well if you remember the radius is equal to half of the diameter. So we knew that the diameter was 10, and we divide that by 2, and the radius is going to be equal to 5. Okay now we want to really talk about circle equations. And the standard form is x minus h quantity squared plus y minus k quantity squared equal to r squared, and hk is your center, and the key thing to remember is you're always going to be subtracting in here, so whatever h and k are you want to subtract those things. So find the standard form of the equation of a circle at the center of 0, 0, and a radius of 6. So that means that h is 0, and k is 0, and then r is 6. So looking up at our formula we have x minus h, so x minus 0 quantity squared or just x squared, plus y minus 0 or just y squared equal to 6 squared, but we would say equal to 36, that would be the equation of that circle. And then find the standard formula equation with a circle of 3 negative 8 and a radius of 9, so now we have h is equal to 3, so that gives us x minus 3 quantity squared and k is going to be equal to negative 8, so that will be plus y minus a negative, so we're going to add 8 quantity squared, and then the r was equal to 9, but we have to square that remember, so it's equal to 81. And we want to find the standard form of a center with negative 2, 3, and a diameter. So if the diameter is 10, that means that our radius is going to be half of that, which we already said was 5. So now we know r, I'll just write it over here, h is negative 2, so we have x plus 2, it's always the opposite sign, plus k is 3, so it's y minus 3 quantity squared, and if r is 5, that means that r squared is 25. Now we're going to use a graph to help us figure some of these things out, so we want to know what the center of the circle is, well if you look at it, it looks like it's about right here, in fact let's make it a nice pretty color, it looks like it's about right here, but we want to verify that we really know that that's what it is. So we have the point here, looks like negative 5 and 3, and the point over here looks like it's 1, negative 1, well the midpoint would be the center of the circle, so if we take the midpoint, we would have negative 5 plus 1, remember you add for the midpoint, divided by 2, and then for the y's, we have 3 minus 1 because we're adding a negative over 2, I don't know why my lines are so squiggly, so negative 5 minus plus 1 would be negative 4 over 2, and then 3 minus 1 would be 2 over 2, so we know that the center of our circle is going to be negative 2, 1. Now we need to find the radius, well we could find the distance, now that we know what this point is, and look it was negative 2, 1, once we find that point, we could just find the distance between that and one of the points on the circle and that would give us the radius, so the distance I'm going to use negative 2, 1, okay so the square root of again negative 2 minus 1, subtract the x's and square them, and then 1 plus 1, subtract the y's and square them, so that gives me negative 3 squared which is going to be 9, and 2 squared which is going to be 4, so the radius is the square root of 13, and now they want me to write the standard formula equation, let's do that over here, standard formula equation, use the center, this is h is equal to negative 2, negative 2, k is equal to 1, and r is equal to the square root of 13 which means that r squared is going to be equal to, squaring the square root cancels it out, we just have 13, so now we're ready to write it, x plus 2, since it was a minus 2, quantity squared, plus y minus 1, quantity squared is equal to 13.