Theorem : Distance Formula


   
 
Distance Formula
Theorem
 
The length of the line segment AB, which joins A (x1, y1) and B (x2, y2) is given by
 
 
 
Proof
 
Let A (x1, y1) and B (x2, y2) be two points in the plane.
 
Let d = distance between the points A and B.
 
Draw AL and BM perpendicular to x-axis (parallel to y-axis).
 
Draw AC perpendicular to BM to cut BM at C.
 
In the figure,
 
OL = x1, OM = x2 [AC = LM = OM - OL = x2 - x1]
 
MB = y2, MC = LA = y1 [CB = MB - MC = y2 - y1]
 
From the right-angled DACB,
 
 
 
 
 
Note:
 
i) If the points A and B lie on the x-axis, then the ordinates of A and B are zeros.
 
i.e., A (x1, 0), B (x2,0)
 
 
ii) If the points A and B lie on the y-axis, then the abscissae of A and B are zeros.
 
i.e., A (0,y1) and B (0,y2)
 
 
iii) Distance of any point A (x, y) from the origin
 
 
Example:
 
Find the distance between the following pair of points:
 
A (1,2) and B (4,5).
 
Suggested answer:
 
Using the distance formula, we have
 
 
 
 
 
     
   
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