# EXAMPLE PROBLEMS ON SECTION FORMULA

Example Problems on Section Formula :

We use the section formula to find the point which divides the line segment in a given ratio.

The point P which divides the line segment joining the two points A (x1,  y1) and B (x2, y2) internally in the ratio l : m is If P divides a line segment AB joining the two points A (x1,  y1) and B (x2, y2) externally in the ratio l : m is, ## Example Problems on Section Formula

Example 1 :

Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2 : 3.

Solution : Here x1  =  -1, y1  =  7, x2  =  4 , y2  =  -3  m = 2  and n = 3

=  [2 (4) + 3 (-1)]/(2+3) , [2 (-3) + 3 (7)]/(2+3)

=  (8 - 3)/5, (-6+21)/5

=  5/5, 15/5

=  (1, 3)

Example 2 :

Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

Solution : Let "C" and "D" be the points which divides the line segment into three equal parts.

length of AC = 1 unit

length of CD = 1 unit

length of DB = 1 unit

So C divides the line segment in the ratio 1 : 2

Here x1  =  4, y1  =  -1, x2  =  -2, y2  =  -3  m  =  1  and n = 2

=  [1(-2) + 2(4)]/(1+2) , [1(-3) + 2(-1)]/(1+2)

=  (-2 + 8)/3, (-3-2)/3

=  6/3, -5/3

=  (2,-5/3)

So, D divides the line segment in the ratio 2 : 1

Here x1  =  4, y1  =  -1, x2  =  -2 , y2  =  -3  m = 2  and n = 1

=  [2 (-2) + 1 (4)]/(2+1) , [2 (-3) + 1 (-1)]/(2+1)

=  (-4 + 4)/3, (-6-1)/3

=  0/3, -7/3

=  (0,-7/3)

Example 3 :

To conduct sports day activity, in your rectangular shaped school ground ABCD, lines have been drawn which chalk powder at a distance of 1 m each 100 flower pots have been placed at a distance of 1 m from each other along AD,as shown in figure given below. Niharika runs ¼ th the distance AD on the second line and posts a green flag. Preet runs 1/5th the distance AD on the eight line and posts a red flag.

What  is between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags,where should she post her flag?

Solution :

It can be observed that Naharika posted a green flag at 1/4 th of the distance AD

That is,

(1/4) x 100  = 25 m

from the starting point of the 2nd line.

So, the coordinate of the point is (2, 25)

Preet post red flag at 1/5th of the distance AD.

(1/5) x 100  =  20m

from the starting point of the 8th line.

So, the coordinates of this point R are (8, 20). Distance between these flags.

=  √(x2 - x1)2(y2 - y1)2

=  √(8 - 2)2 + (20 - 25)2

=  √62 + (-5)2

=  √36 + 25

=  √61 m

The point at which Rashmi should post her blue flag is the midpoint of the line joining these points.

x  =  (8 + 2)/2, y  =  (25 + 20)/2

=  10/2, 45/2

=  (5, 22.5)

So, Rashmi should post her flag at 22.5 m on 5th line. After having gone through the stuff given above, we hope that the students would have understood, example problems on section formula.

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