## HOW TO SHOW THE GIVEN POINTS ARE COLLINEAR USING SECTION FORMULA

How to Show the Given Points are Collinear Using Section Formula ?

If three points are collinear, then one of the points divide the line segment joining the other two points in the ratio r : 1.

The section formula can be used only when the given three points are collinear.

Example 1 :

Using section formula, show that the points A (7, −5), B (9, −3) and C (13, 1) are collinear.

Solution :

Section formula  =  (mx2+nx1) / (m+n),  (my2+ny1) / (m+n)

Let the point B divides the line segment joining the point A and C in the ratio k : 1

k (13) + 1 (7) / (k + 1), k (1) + 1 (-5) / (k + 1)  =  (9, -3)

(13k + 7)/(k + 1), (k - 5) / (k + 1)  =  (9, -3)

Equating the x and y coordinates

 (13k + 7) / (k + 1)  =  9 (13k + 7)  =  9(k + 1)13k + 7 = 9k + 913k - 9k  =  9 - 74k  =  2 k = 2/4 k = 1/2 (k - 5) / (k + 1)  =  -3k - 5  =  -3(k + 1)k - 5  =  -3k - 3k + 3k  =  -3 + 5 4k  =  2k  =  2/4k  =  1/2

We get the same values for k, hence the point A, B and C are collinear.

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