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# Coordinate plane word problem

Video transcript

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.