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CCSS.Math:

a circle is centered at the point C which has the coordinates negative 1 comma negative 3 and has a radius of 6 where does the point P which has the coordinates negative 6 comma negative 6 lie and we have three options inside the circle on the circle or outside the circle and the key realization here is just what a circle is all about if we have the point C which is the center of a circle a circle of radius 6 so let me draw that radius so let's say that is its radius it is 6 units the circle will look something like this remember the circle is the set of all points that are exactly 6 units away from that Center so that's that's the definition of a circle it's a set of all points that are exactly 6 units away from the center so if for example P is less than 6 units away it's going to be inside the circle if it's exactly 6 units away it's going to be on the circle and if it's more than 6 units away it's going to be outside of the circle so the key is is let's find the distance between these two points if the distance is less than 6 inside distance equals 6 we're on the circle distance more than 6 we are outside of the circle so let's do that so if we wanted to find and there's different notations for the distance well I'll just write D or I could write the distance between C and P is going to be equal to and the distance formula comes straight out of the Pythagorean theorem but it's going to be the square root of our change in x squared plus our change in Y squared so what is our change in X so our change in X if we view C as our starting point and P as our end point but we could do it either way our change in X our change in X is negative 6 minus negative 1 so negative 6 minus negative 1 and we're going to square it so what we have inside here that is change in X so we're taking our change in x squared and then plus our change in Y squared so we are going we're going from negative 3 to negative 6 so our change in Y is negative 6 minus negative three negative six minus negative three and we're going to square everything so that is our change in Y inside the parentheses and we're going to square it so this is equal to this is equal to negative six negative six plus positive one is one way to think about it so this is negative five squared and then this is negative six plus three so plus negative three squared and once again you can see our change in X is negative five we go five lower in X and we're going five three lower in Y so our change in Y is negative three so this is equal to the square root of twenty-five square root of 25 plus nine square root of 25 plus nine which is equal to the square root of 34 now the key is is the square root of 34 less than six greater than six or equal to six well we know that six is equal to the square root of 36 so the square root of 34 is less than the square root of 36 so I could write the square root of 34 is less than the square root of 36 and so the square root of 34 is less than six these are square root of 36 is 6 and so since the distance between C and P is less than 6 we are going to be on the inside of the circle if I somehow got square root of 36 here then we'd be on the circle and if I somehow got square root of 37 here or something larger we would have been outside the circle