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## High school geometry

### Course: High school geometry > Unit 6

Lesson 3: Problem solving with distance on the coordinate plane- Area of trapezoid on the coordinate plane
- Area & perimeter on the coordinate plane
- Points inside/outside/on a circle
- Points inside/outside/on a circle
- Challenge problem: Points on two circles
- Coordinate plane word problem
- Coordinate plane word problems: polygons

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# Coordinate plane word problem

CCSS.Math:

Watch Sal solve an example where has to determine which of the Minions a wizard can reach using the coordinate plane. Created by Sal Khan.

## Want to join the conversation?

- why is it 3 square roots of 2 instead of the square root for 18?(13 votes)
- Sal was just re-writing it in a way that makes it a little more obvious what the decimal value of the number would be.
`3√2`

is clearly greater than`3`

but less than`6`

. However, for the purpose of the original question (as you can see near the end of the video) it's actually easier to leave the result as`√18`

- that makes it easier to compare to the`√36`

spell length and to the other minion distances.(16 votes)

- Why does Sal put Absolute Value signs around the change in x and y? Isn't distance always positive? (Around1:50-2:00)(7 votes)
- Distance is always positive but change in x's and y's can be negative.

E.g. change in x's between A(4;9) and B(1;9) is equal to 1-4=-3, but distance is 3 ( |-3|=3).(15 votes)

- how is sqrt(9 +9) = 3sqrt(2)?(5 votes)
- Use the fact that 9+9 = 2*9

Sqrt(9+9) = sqrt(2*9) = sqrt(2) * sqrt(9) = 3*sqrt(2) , and this is the most you can simplify the radical because 2 has no repeated factors except for 1.(14 votes)

- At1:24what is the little number going down?(7 votes)
- Subscript to indicate that which point it belongs to. In the example we had 2 points with the (x,y) coordinates. If we labelled them both (x,y) it will be confusing. So we assign either point to be point 1 (usually the first given point) and the other is point 2. So point 1 has the coordinate (x subscript 1, y subscript 1) and point 1 has the coordinate (x subscript 2, y subscript 2)(9 votes)

- Why doesn't he just use the distance formula?(3 votes)
- The distance formula is basically the same thing as the Pythagorean Theorem so he can use the distance formula or the Pythagorean Theorem. It doesn't matter at all. Also, as you may have noticed, Sal isn't very fond of memorizing formulas :)

However, he basically DOES basically use the distance formula.(7 votes)

- if (b,b-1),(b+2,b+1) and (b,b+3) are three consecutive points of a square what's its area?(2 votes)
- Since we know it is a square, we only need to know the length or distance of one side to find the area. To find the distance we use Pythagorean theorem , or d=√((change in x)²+(change in y)²) or d=√((y₂-y₁)²+(x₂-x₁)²)

Point 1 (x₁,y₁) given (b, b-1) and point 2 (x₂,y₂) given (b+2,b+1)

We can identify x₁=b and x₂=b+2, therefore (x₂-x₁)²=(b+2-b)²=2²=4

And y₁=b-1, y₂=b+1, so (y₂-y₁)²=((b+1)-(b-1))²=(b+1-b+1)²=2²=4

So d=√((y₂-y₁)²+(x₂-x₁)²)=√(4+4)=√8

Area of the square is A=S²=(√8)²=8 (unit)²(5 votes)

- At1:37, what does the triangle thing represent?(1 vote)
- The triangle thing is the greek letter "delta", and it can be used to denote "change in" when you're doing math or science.(6 votes)

- I know this is a bit off topic but what grade would you learn to find the area of a triangle with given vertices.(2 votes)
- Area of the triangle with angles at the vertices? I'm not sure what you're talking about.

I'll assume you're talking about geometry which is a 10th grade subject, to trig which is taken after algebra 2.(2 votes)

- witch point is longer than 6 meters?(2 votes)
- None of them are further than 6m(2 votes)

- Why didn't Sal mark d(P1,P4) = 4 ⋅ √ ̅2 (he marked d(P1,P2) = 3 ⋅√ ̅2) ?

As goes with any 45,45,90 triangle, A=B=C√ ̅2.(2 votes)

## 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.