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# i needs help

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i needs help

what is the distance formula?

Aug 3, 2017

#2
+7347
+3

Let's look at the points  (4, 5)  and  (6, 2)  for example.

We can make a right triangle out of these points like this...

Notice that side  c  is the distance between the two points.

And..we can find the length of side  c  using the Pythagorean theorem, which says..

a2    +     b2     =  c2

The length of side  a  =  5 - 2  =  3

(5 - 2)2 +     b2     =  c2

The length of side  b  =  6 - 4  =  2

(5 - 2)2 + (6 - 4)2  =  c2

Take the positive square root of both sides.

$$\sqrt{(5-2)^2+(6-4)^2}=c$$

And this is where the distance formula comes from. It says..

$$\text{distance}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$

And, as CPhill noted, the order of subtraction within the parenthesees doesn't matter.

Aug 3, 2017

#1
+94586
+2

Suppose that we have two points (x1, y1)  and (x2, y2)

The distance between these is just

sqrt [  (  x2 - x1)^2 + ( y2 - y1)^2 ]   ..  actually....the order of subtraction is immaterial...we could subtract (x1 - x2 )  or (y1 - y2) or both in this order

Example.....distance between (5,3)  and (8, 2)

sqrt [  ( 5 - 8)^2 + (3 - 2)^2 ]   =  sqrt [ (-3)^2 + 1^2 ]  =  sqrt [ 9 + 1 ] =  sqrt (10)  units ≈  3.16 units

Aug 3, 2017
edited by CPhill  Aug 3, 2017
edited by CPhill  Aug 3, 2017
edited by CPhill  Aug 3, 2017
#2
+7347
+3

Let's look at the points  (4, 5)  and  (6, 2)  for example.

We can make a right triangle out of these points like this...

Notice that side  c  is the distance between the two points.

And..we can find the length of side  c  using the Pythagorean theorem, which says..

a2    +     b2     =  c2

The length of side  a  =  5 - 2  =  3

(5 - 2)2 +     b2     =  c2

The length of side  b  =  6 - 4  =  2

(5 - 2)2 + (6 - 4)2  =  c2

Take the positive square root of both sides.

$$\sqrt{(5-2)^2+(6-4)^2}=c$$

And this is where the distance formula comes from. It says..

$$\text{distance}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$

And, as CPhill noted, the order of subtraction within the parenthesees doesn't matter.

hectictar Aug 3, 2017