Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. We're going to look at the distance formula, which is based on two given points. Let's put a point here and call it X2Y2. And let's put another point maybe down here and call it X1Y1. Now the distance formula is really, really established off of a right triangle using the Pythagorean theorem. If we look at this right triangle and we want the distance of this line here, we can find it by using our concept of a ^2 + b ^2 = C ^2. The distance here is really found by just taking the farthest to the right point and subtracting the furthest to the left. Because it's a horizontal line, we just talk about the furthest right minus the furthest left X 2 -, X one. I'm going to do the same thing here. The highest coordinate minus the lowest. And because it's a vertical line, we're just going to look at the YSY 2 -, y one. So if we look at this, we'd get our A value as X2 minus X1 squared and our B value as Y 2 -, y one squared equaling Rd. squared. Well, how do we get rid of a square? We square root it and this is where the distance formula originates from, or one way to find the distance formula. Another important formula for us this semester is going to be the midpoint formula. And the midpoint formula states that if I know two points, the midpoint is the average of the X's. We take the 2X values, divide by two, and the two Y values added together and divided by two. That's how we find our midpoint. Have a great day. A circle is the set of all points and a Cartesian coordinate system that are in equal distance, called the radius from a fixed point called the center. Let's look at that graphically. If we have a center point, let's call it H, K, and every single point on the circle is an equal distance away called the radius. So any point that I choose, let's call this point X, Y, every single point should be this distance away. If we thought about dropping and making a right triangle by dropping a vertical line and a horizontal line, this new point went over the same distance as that XY. So this new point would be called X and we went up the same amount as our center or K Now if we use the distance formula, the distance formula says that we are going to subtract the X coordinates here. So this distance here, because it's a horizontal line, is just going to be my X minus my H We don't worry about the YS in that particular case because the YS were the same. If we look at this vertical line, it's going to be the upper Y minus the lower Y, or in this case, y -, K, and we square them up both. So this is really the distance formula. We're finding the distance between two points. Well, it actually is also the formula for a circle. If I square each side and change this D to R for radius, R-squared equals the quantity X -, H ^2 + y -, K ^2 where H, K is the center of the circle and R is the radius. Now that's a very important form of a circle, but there's another one that's also important. So the standard form is the one we just looked at. X -, H quantity squared plus y -, K quantity squared equal R-squared. If we took this and used our wonderful algebra skills and foiled it out, we'd get a general form that says X ^2 + y ^2 plus CX plus DY plus e = 0. Now eventually with quadratics, we actually put coefficients here. So we'd have AX squared plus B y ^2 plus CX plus D y + e = 0. And for this second more generic equation to be a circle, a has to equal B. Unit circle means that our radius is one. So in a unit circle we have a radius of one and the center is at the origin. So if we have the center at the origin, the origin is the .00 X -0 ^2 + y - 0 ^2 equal 1 ^2. So a unit circle is just X ^2 + y ^2 equal 1. Thank you and have a wonderful day. This is Anna Cox.