**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 (x_{1}, y_{1}) and B (x_{2}, y_{2}) internally in the ratio l : m is

If P divides a line segment AB joining the two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) externally in the ratio l : m is,

**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 x_{1} = -1, y_{1} = 7, x_{2} = 4 , y_{2} = -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 x_{1} = 4, y_{1} = -1, x_{2} = -2, y_{2} = -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 x_{1} = 4, y_{1} = -1, x_{2} = -2 , y_{2} = -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.

= √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

= √(8 - 2)^{2} + (20 - 25)^{2}

= √6^{2} + (-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 5^{th} 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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