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The Section Formula:

When you have a straight line in front of you and you are thinking how to divide this line in a specific ratio that is when section formula comes to your rescue. When a point A on a line segment divides it in a ration m:n, we have the liberty to use the section formula to find the coordinates as well as the distance between the points. This need not be solely internal. A point can lie outside of the line segment as well and a section formula can help us in finding the coordinates both internally and externally. The section formula is one of the most important and basic fundamentals of coordinate geometry. Read on to learn this concept!

Image Credits: brilliant.org

For example, AB is a line with coordinates (x1,y1) and (x2,y2). Point P divides the line segment AB in the ration m:n, then the coordinates of P is
x = ( mx2+nx1 )/ ( m+n )

Y = ( my2 + ny1 )/ ( m+n )

This is the case of internal division.

Now when the point P lies outside the line segment AB, then the formula that needs to be used for computation is different.

Image Credits: brilliant.org

Here, the point P lies externally. Hence, the formula to compute the coorfdinates in this case will be:
x = ( mx2-nx1 )/ ( m-n )

Y = ( my2 – ny1 )/ ( m-n )

In case the point P lies exactly in the middle of the line segment AB,  we take the value of m to be 1 and that of n to be 1 and substitute these values in the formula for internal division. On substitution we get,

x = ( mx2+nx1 )/ ( m+n )

or, x = ( 1x2+1x1 )/ ( 1+1 )

or, x = ( x2+x1 )/ ( 2 )

 

Y = ( my2 + ny1 )/ ( m+n )

Or, Y = ( 1y2 + 1y1 )/ ( 1+1 )

or, Y = ( y2 + y1 )/ ( 2 )

 

Hence, coordinates of point P is ( x2+x1 / 2 , y2 + y1 / 2 ). This is also known as the midpoint theorem.

Some worked out examples for section formula

  1. Find the coordinates of the point that divides the line with coordinates (5,5) and (-4,-4) in the ratio 3:2, internally.

Answer: Let P (x,y) be the point on the line segment that divides it in the ratio 3:2.

Thus, the coordinates will be

x = ( mx2+nx1 )/ ( m+n )

Y = ( my2 + ny1 )/ ( m+n )

Where m=3, n=2, x2=-4, y2= -4, x1=5 and y1=5

Substituting we get,

x = ( 3 . (-4) + 2 . (5) ) / ( 3+2 )

Y = ( 3 . (-4) + 2(5) ) / (3+2 )

Or, x = -2/5

Y=-2/5

 

  1. The vertices of a parallelogram are P(-2,3), Q(3,-1), R(a,b) and S(-1,9). Find the value of a and b.

Answer: The diagonal of a parallelogram bisects each other. Also, the coordinates of the mid points od PR will be the same as that of QS.
Using the mid point formula we can get the value of the midpoint of the line segment QS.

It is, ( x2+x1 / 2 , y2 + y1 / 2 ).
It is, ( -1+3 / 2 , 9 + (-1) / 2 ).

(1, 4)

 

This is the same as the coordinates of the midpoint of PR.

Or it is equal to ( a+ (-2) / 2 , b + 3 / 2 ).

On setting them as equal we get

(a-2)/2 = 1 and (b+3)/2 = 4

a = 4 and b= 5

 

  1. Find the coordinates of the point that trisects the line joining (1, -2) and (-3 , 4)

Answer: Let P = (1, -2) and Q = (-3, 4) be the points.
Let the points of trisection be A and B.

Then, PA = AB = BQ = x
AQ = AB + BQ = 2x and PB = PA + AB = 2x
PA : AQ = x : 2x = 1 : 2
and PB : BQ = 2x : x = 2 : 1

So, A divides PQ internally in 1 : 2 ratio and B divides it internally in 2 : 1 ratio

Thus the coordinates of A and B are:

A = ({ 1 x (-3) + 2 x 1/ 1+ 2 }, { 1 x 4 + 2 x (-2) / 1 + 2}) = (-1/3 , 0)
B = ({ 2 X (-3) + 1 x 1 /  2 + 1}, { 2 x 4 + 1 x (-2)/ 2 + 1}) = (-5/3, 2)

Hence the points of trisection are (-1/3,0) and (-5/3,2)

  1. If point A ( -1,2 ) divides the line segment joining B (2,5) and C internally, find the coordinates of the point C.

Answer: Let the coordinates of C be (x.y) it is given that BA : AC = 3:4. So the coordinates of  A are
( 3x + 4.2 / 3 + 4 , 3y + 4.5 / 3+4) = (3x+8/7 , 3y+20/7)

According to the sum, these coordinates are (-1,2)
Hence, 3x+/7 = -1 and 3y + 20/7 = 2
on solving for x and y we get
x = -5 and y = -2
therefore C is (-5,-2)

  1. If the points (1,2), (-4,-4), (9,4) and (p,3) are the vertices of a parallelogram, find the value of p.

Answer: Use the mid point formula for the diagonal for (1,2) and (9,4) and use it with (-4,-4) to get the value of p.

 

Thus, to summarise,

  1. The coordinates of the point dividing the line segment joining (x1, y1) and (x2, y2) in the ratio m : n internally is given by (mx2 + nx1 )/  (m + n), (my2 + ny1) / (m + n)
  2. The coordinates of the point dividing the line segment joining (x1, y1) and (x2, y2) in the ratio m : n externally is given by  (mx2 – nx1 )/  (m – n), (my2 – ny1) / (m – n)

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