9.4 Distances and Distance Formula
With the help of x and y axis we saw how the position of a point in the coordinate plane was determined. We shall now extend this theory to calculate the distances between any two points in the coordinate plane.
Let A (x, y) and B (x_{2}, y_{2}) be two points in the coorrdinate plane as shown below :
Figure 9.4
In order to find the distance between points A and B we go through following steps and use the distance formula.
Step 1 Draw a line parallel to xaxis through the point A and draw a line parallel to yaxis through point B such that they intersect at point C.
Step 2 We now have a right triangle with seg AB as the hypotenuse. Therefore, by Pythagorean theorem,
(AB)2 = (AC)2 + (BC)2
\
AB =
Step 3 The coordinates of point C can be determined as (x_{2}, y_{1}).
Step 4 The distance between A and C is,
AC = (x_{2}, x_{1})
\
(AC)^{2} = (x_{2}, x_{1})^{2} and
The distance between B & C is
BC = (y_{2}, y_{1})
\
(BC)^{2} = (y_{2}, y_{1})^{2}
Step 5 Substituting the values of (AC)^{2} and
(BC)^{2} in
eg (1) we get
This is the distance formula.
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