Coordinate plane word problem

alyssa is playing a video game her character is on a quest to vanquish an evil sorcerer and his minions from the land her character is a wizard who spells have a range of 6 meters the locations of objects in the game are stored by the computer in terms of x and y-coordinates so 5 comma 4 is the location of Alyssa's wizard 8 , 7 is the location of minion a 2 comma negative 1 is the location of minion B 9 common comma 0 is location of minion C so what I wanted to do and I want you to pause this video and I want to think about given that her wizard has a range of 6 meters which of these minions can the wizard actually reach I'm assuming you've given a go at it we just have to remind ourselves we're to figure out which of these minions are in reach we to say well which of these points are within 6 units we're assuming that these units are in meters right over here which of these points are within six units of 5 comma 4 and to think about that we just have to calculate the distance between this point and this point that point and that point that point in that point and see if they are greater than or less than 6 meters and how do we calculate a distance between two points well if this is some point right over here that's x1 comma y1 and then this is another point right over here x2 comma y2 and we want to calculate this distance right over here the distance formula comes straight out of the Pythagorean theorem it's the Pythagorean theorem tells us if this side right over here is our change in Y and that let's actually just write that as the absolute value of our change in Y and let's say that this side right over here is the absolute value of our change in X absolute value of our change in X the fact Pythagorean theorem tells us that this one the hypotenuse is going to be the square root of the sum of the squares of the two sides so change in x squared plus change in Y squared plus change in Y squared you might say hey what happened to the absolute value well when I square it it's going to be positive anyway so I don't have to write down the absolute value so really I just need to figure out between each of these two points what is the change in X what's the change in Y square them add them together and then take the square root so for example let's find let's find so if I were to call this if I were to call this pp1 if I were to call this p2 we call this p3 or do them in different colors so you can keep track of what I'm doing this is p3 and let's say this is p4 so let's first think about the distance the distance between p1 and p2 and p2 well that's going to be equal to the square root of our change in x squared so our change in X is 3 that squared is 9 plus our change in Y squared our change in Y is also 3 that squared is 9 so this is going to be square root of 18 which is the same thing as 3 square roots of 2 3 square roots of 2 now is this more or less than 6 well 3 times 2 is equal to 6 square root of 2 is less than 2 it's one point something it's 1 so this right over here is going to be this over here is going to be less than 6 so p2 is in range Alissa's Wizard can get minion a minion a she can attack now let's think about minion B so the distance between P 1 and P 3 and P 3 is going to be equal to the square root of so your change in X it's negative 3 negative 3 squared is positive 9 our change in Y to go from 4 to negative 1 it's negative 5 that squared is 25 so 9 plus 25 which is equal to the square root of 34 now is this greater than or equal to 6 well 34 30 square root of 36 is 6 so this is going to this is a square root of a lower number so this is going to be less than 6 as well so P minion B is also in reach now let's think about this last point the distance the distance between P 1 and P 4 and P 4 is going to be equal to the square root of our change in x squared change in X is 4 squared is 16 plus our change in Y squared our change in Y is negative 4 but you square that you get another 16 so this is going to be square root of 32 which could be written out actually we could just leave it as square root of 32 square root of 32 is clearly less than the square root of 36 which is 6 so this is also going to be less than 6 so she can get at all of the minions she can get at all of the minions they're all within six units of her now which of these is the furthest away well actually if we were to write this we simplified this but we could write this as a square root of 18 square root of 18 is clearly the smallest out of square root of 18 square root of 32 and square root of 34 so minion minion a is the closest and minion B square root of 34 is the farthest or I should say the farthest is is the farthest