Distance Formula Proof

For distance formula proof statistics will be deals with the co-ordinates parameters.Formula for distance calculation by the co ordinates of the points given. For calculating distance statistics between the points we have the distance formula proof and we substitute the points in that formula we find the distance between the points.In this article we have the distance formula proof and the problems using the distance formula.
Distance Formula Proof:
Distance between two points co ordinates is a basic concept in geometry.Now, we give an algebraic expression for the same.

Let P1  (x1, y1) and P2 (x2, y2) be two points in a Cartesian plane and denotes the distance between P1 and P2 by d(P1, P2) or  by  P1P2. Draw the line segment ` bar(P_1P_2)`

distance formula proof

The segment ` bar(P_1P_2)` is parallel to the x axis  Then y1 = y2. Draw P1 L and P2 M, perpendicular to the x-axis. Then d(P1,P2) is equal to the distance between L and M. But L is (x1, 0) and M is (x2, 0).

So the length LM = |x1-x2| Hence d (P1, P2) = |x1-x2|.

therefore, [d(P1,P2)]2= |x1-x2|2+ |y1-y2|2

=(x1-x2)2+(y1-y2)2

=(x2-x1)2+(y2-y1)2
d(P1,P2) = `sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

Problems Using Distance Formula Proof:
Example 1:
Find the distance between the points A(-2,3) and B(3,7).

Solution:
Assume d be the distance between A and B.             (x1,y1) = (-2,3)

Then d (A, B) =`sqrt((x_2-x_1)^2+(y_2-y_1)^2) `                   (x2,y2) =  (3,7).

=`sqrt((3+2)^2 +(7-3)^2)`

=`sqrt(5^2+4^2)`

=`sqrt(25+16)`

=`sqrt41`
Example 2:
Find the distance between the points A(-1,1) and B (3,3).

Solution:
Assume d be the distance between A and B.                (x1,y1)= (-1,1)

Then d (A, B) =`sqrt((x_2-x_1)^2+(y_2-y_1)^2) `                                (x2-y2)= (3,3).

=`sqrt((3+1)^2 +(3-1)^2)`

=`sqrt(4^2+(2)^2)`

=`sqrt(16+4)`

=`sqrt20`
=2`sqrt 5`

I am planning to write more post on Equivalence Relations. Keep checking my blog.

Example 3:
Find the co ordinates distance between the points A(-1,5) and B (1, 4).

Solution:
Assume d be the distance between A and B.                       (x1,y1)= (-1,5)

Then d (A, B) = `sqrt((x_2-x_1)^2+(y_2-y_1)^2) `                          (x2-y2)= (1,4).

=`sqrt((1+1)^2 +(4-5)^2)`

= `sqrt(2^2+(-1)^2)`

=`sqrt(4+1)`
=`sqrt5`