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CCSS.Math:

so we have a circle here and they specified some points for us this little orange issue Argos maroonish red point right over here is the circle is the center of the circle and then this blue point is a point that happens to sit on the circle and so with that information I want you to pause the video and see if you can figure out the equation for this circle all right let's work through this together so let's first think about the center of the circle and the center of the circle is just going to be the coordinates of that point so the x-coordinate is negative 1 and then the y-coordinate is 1 so Center is negative 1 comma 1 and now let's think about what the radius of the circle is well the radius is going to be the distance between the center and any point on the circle so for example for example this distance the distance of that line let's see I could do a thicker thicker version of that just this this line right over there something strange about my something strange about my pen tools making that very thin let me do it one more time okay that's better the distance of that line right over there that is going to be the radius so how can we figure that out well we can set up a right triangle essentially use the distance formula which comes from the Pythagorean theorem to figure out the length of that line so this is the radius we could figure out a change in X so if we look at our change in X right over here our change in X as we go from the center to this point so this is our change in X and then we could say that this is our change in Y that right over there is our change in Y and so our change in x squared plus our change in Y squared is going to be our radius squared that comes straight out of the Pythagorean theorem this is a right triangle and so we can say that R squared is going to be equal to our change in x squared plus our change in Y square plus our change in Y squared now what is our change in X square or what is our change in X going to be our change in X is going to be equal to well when we go from the radius to this point over here our X goes from negative 1 to 6 so you can view it as our ending X minus our starting X so negative 1 minus negative sorry 6 minus negative 1 is equal to 7 so let me so we have our change in X this right over here is equal to 7 if we viewed this is the start point and this is the end point it would be negative 7 but we really care about the absolute value of the change in X ended once you square it it all becomes the positive anyway so our change in X right over here is going to be positive 7 and our change in Y well we are starting at we are starting at Y is equal to 1 and we are going to Y is equal to negative 4 so it would be negative 4 minus 1 which is equal to negative 5 and so our change in Y is negative 5 you could view this distance right over here is the absolute value of our change in Y which of course would be the absolute value of 5 but once you square it it doesn't matter the negative sign goes away and so this is going to simplify to 7 squared change in x squared is 49 change in Y squared negative 5 squared is 25 so we get R squared we get R squared is equal to 49 plus 25 so what's 49 plus 25 let's see that's going to be 54 was it 74 R squared is equal to 74 did I do that right yep 74 and so now we can write the equation for the circle the circle is going to be all of the points that are well and Ike let me write the so if R squared is equal to 74 R is equal to the square root of 74 and so the equation of circle is going to be all points X comma Y that are this far away from the center and so what are those points going to be well the distance is going to be X minus the x-coordinate of the center X minus negative one squared let me do that in blue color minus negative one squared plus y minus y minus the y-coordinate of the center Y minus one squared squared is equal is going to be equal to R squared is going to be equal to the length of the radius squared well R squared we already know is going to be 74 74 and then if we want to simplify a little bit you subtract a negative this becomes a positive so it simplifies to X plus one squared plus y minus one squared is equal to 74 is equal to 74 and we are all done