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 whose 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 comma 7 is the location of Minion A. 2 comma negative 1 is the location of Minion B. 9 comma 0 is the location of Minion C. So what I want to do-- and I want you to pause this video. And I want you 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're given a go at it. And we just have to remind ourselves, to figure out which of these minions are in reach, we have 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 6 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 and 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. The Pythagorean theorem tells us, if this side right over here is our change in y-- and 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, the 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. 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, if I were to call this P1, if I were to call this P2, maybe we call this P3-- I want to 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 between P1 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. 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 1 point something. So this right 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 think about Minion B. So the distance between P1 and P3 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, the square root of 36 is 6. So this is the square root of a lower number. So this is going to be less than 6 as well. So Minion B is also in reach. Now let's think about this last point. The distance between P1 and P4 is going to be equal to the square root of our change in x squared-- the 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, and you get another 16. So this is going to be square root of 32, which 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. They're all within 6 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 the 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 A is the closest. And Minion B, square root of 34, is the farthest.