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# Challenge problem: Points on two circles

CCSS.Math:

## Video transcript

point a is that negative five comma five so this is negative five right over here this is one two three four five that's five right over there so point a is right about there so that is point a just like that at negative five comma five and then it's a center of circle a which I won't draw just yet cuz I don't know the radius of circle a point B is that let me sir let me underline these in the appropriate color point B is it 3 comma 1 so 1 2 3 comma 1 so that's point B right over there it's a center of circle B Point P is at 0 0 so it's right over there at the origin and it is on circles a and B well that's a big piece of information because that tells us if this is on if this is on both circles then that means that this is b's radius away from the set from the point b from the center and this tells us that it is circle a's radius away from its center which is at Point a so let's figure out what those radii actually are and so we can imagine let me draw let me draw a radius or the radius for circle a we now know since P sits on it that this could be considered the radius for circle a the radius for circle a and you could use the distance formula but we'll see that the distance formula is really just falling out of the Pythagorean theorem so the distance formula tells us the radius right over here this is just the distance between those two points so the radius or the distance between those two points squared is going to be equal to our change in X values between a and P so our change in X values we could write it as negative 5 minus zero squared negative 5 minus zero squared that's our change in X that's our change in X negative five minus zero squared plus our change in Y 5 minus 0 5 minus 0 squared which gets us that are our distance between these two points which is the length of the radius school is equal to negative five squared negative five squared plus five squared or we could say that the radius is equal to the square root of this is 25 this is 25 5050 we can write as 25 times 2 so this is equal to the square root of 25 times the square root of 2 which is 5 times the square root of 2 so this distance right over here is 5 times the square root of 2 now I said this is just the same thing as the Pythagorean theorem why well if we were to construct a right triangle right over here if we were to construct a right triangle right over here then we can look at this distance this distance would be the absolute value of negative 5 minus 0 or you could say 0 minus negative 5 this distance right over here is 5 this distance is 2 the distance between 0 & 5 in the Y Direction that's 5 Pythagorean theorem tells us that 5 squared which is 25 plus 5 squared another 25 is going to be equal to your hypotenuse squared and that's exactly what we have here now you might be saying wait wait wait this thing had a negative 5 squared here while here you had a positive 5 but the reason why we could do this is when you square it the negative disappears the distance formula that you could write it this way where you're taking the absolute value and then it becomes very clear that this really is just the Pythagorean theorem this would be 5 squared plus 5 squared 5 squared plus 5 squared the reason why you don't have to do this is because the sign doesn't matter when you square it it always it will become a positive value but either way we figured out this radius now let's figure out the radius of circle be the radius of circle be the radius of circle be same exact idea the radius of circle be squared is equal to our change in X so go from from if we could be we could write it as 3 minus 0 or 0 minus 3 but we'll keep we'll just write it's 3 minus 0 3 minus 0 squared plus 1 minus 0 squared or the radius or the distance between these two points is equal to the square root let's see this is 3 squared plus 1 squared just 9 plus 1 this is the square root 10 the radius of B is the square root of 10 now they asked us which of the following points are on circle a circle B or both circles so all we have to do now is look at these points if they are square root of if this point is the square root of 10 away from point B then it's on the circle it's a radius away a circle is the locus of all points that are that are a radius away from the center if it's 5 square roots of 2 from this point then it's on circle a if it's neither then it's neither or it could be both so let's try these out one by one so Point C is at 4 negative 2 so let me color this in a new color so Point C let me do an orange Point C is that 1 2 3 4 negative 2 Point C is right over there now it looks pretty close just you know this is a hand-drawn drawing so it's not perfect so Point C is there it looks pretty close but let's actually let's actually verify it the distance between Point C and point D so the distance squared is going to be equal to the change in X's so we could say 4 4 minus so we're trying this between C and B it's 4 minus 3 squared 4 minus 3 squared plus negative 2 negative 2 minus 1 squared negative 2 minus 1 squared which is equal to this is 1 squared plus negative 3 squared and so our distance squared is equal to is equal to 10 or our distance is equal to the square root of 10 so this is also the distance right over here is the square root of 10 so this is on the circle if we wanted to draw a circle B it would look something like this and once again I'm hand drawing it so it's not perfect but it would look something I'm going to draw a part of it it looks something like this this is exactly a radius away so let me write this is on Circle Circle B now let's look at this point the point 5 comma 3 so I'll do that in pink so one two three four five comma three so this looks close but let's verify just in case so now our distance is equal to let me just write it this way our distance squared is going to be our change in x squared so five minus three squared five minus three squared plus three minus one squared change in Y three minus one squared and so our distance is going to be equal to actually I want to skip too many steps see this is two squared which is four plus two squared which is another four so our distance is going to be equal to the square root of eight which is the same thing as the square root of two times four which is the same thing as 2 times the square root of 2 square root of four is two and then of course you just have the two left and the radical so this is a different distance away than square root of ten so this one right over here is definitely not on circle B and just eyeballing it you can see that it's not going to be on circle a this distance you're just just eyeballing it is much further much further than five square roots of two and that's also true for Point C Point C is much further than five squared so too you can just look at that you can just look at that visually they're much further than a radius away from a so this point right over here this is neither this is on either circle now finally we have the point negative two comma eight so let me find I'm running out of colors let me see I could use I could use well I guess I'll use yellow again negative two comma eight so that's negative two comma one two three four five six seven eight so it's right it's right over here that is point e just eyeballing it this distance so it's clearly way too far just just looking at it just eyeballing it's clearly more than a radius away from B so this isn't going to be on circle B and also looking at it relative to point a it looks much closer to point a you know it doesn't even seem close then point P is so it looks just inspecting that you could rule this one out but this is going to be neither but we can verify this on our own if we like we can just find the distance between these two points our distance squared is going to be our change in X's so negative two minus negative five negative 2 minus negative five squared plus our change in Y so it's eight minus five squared and so this is our distance squared is going to be equal to negative two minus negative 5 that's negative two plus five so that's going to be three squared plus three squared and you see that right over here Pythagorean theorem this distance right over here is 3 this distance right over here this is your change in X is three change in Y is three three squared plus three squared is going to be the distance squared the hypotenuse squared so our distance squared is going to be or I can say our distance skip a few steps is equal to the square root of we can write this as 9 times 2 or the distance is equal to three times the square root of 2 the radius of circle a is five times the square root of two not three times the square root of two so this is actually going to be inside inside the circle so if we want to draw a circle a it's going to look something like it's going to look something like this it's going to look something like this and point e point e is on the inside point D and point C are on the outside of circle a the only one that sits on any of the circles is point is point C