Section Formula

      Today I'm going to explain section formula using coordinates on Cartesian plane.

But before this, please go through my previous tutorial for basics of Cartesian plane and coordinate geometry i.e.

•  Introduction to co-ordinate geometry

and

• Distance formula and Mid Point Formula

Section Formula

If a point {say (X, Y)} is dividing any line segment [ having points as (x, y) and (x', y')]

in ratio M : N then point (X, Y) can be evaluate by section formula i.e.

X =  [ M x' + N x ]  / [ M+ N ]

Y =  [ M y' + N y ] /  [ M + N ]

with this formula we can have exact location of points lying in between two points and this formula is also used for trisection and four section and onwards.

For an example:-

• Bisection :-

Q- if A (2, 4) and B (-7, 3) and a point P (x, y) is dividing the line into ratio 2 : 3 then find the value of P ???

Sol. for P (x, y) 

x = [ 2×(-7) + 3×2 ] / (2+3)

x = [ -14 + 6] / 5

x = (-8) / 5

x= -8/5

similarly

y= [ 2× 3 + 3 × 4] / (2 + 3)

y= [6 + 12 ] / 5

y= 18 / 5

so point P is 

P ( -8/5, 18/5).

• Trisection:-

If two points P(x, y) and Q (x', y') are trisecting ( points dividing line into three equal parts ) line segment AB with points 

A (6, 5) and B (4, 6)  then these points will be evaluated by method given below :-

{keep in mind on line segment P is lying nearby point A and at left of Q}

for point P ratio will be 1 : 2 

then 

x = [1×4 + 2×6] / (1+2)

x = 16/3

and 

y = [1×6 + 2×5] / (1+2)

y = 16/3

hence point P is 

P (16/3, 16/3)

similarly we can evaluate point Q but this we will take ratio as 2 : 1.

• To find Ratio:-

now this is slightly converse of above problems,

what if ratio is not given???

yes...!

in spite of the ratio is not given there should always be given the respective point for which the ratio is being considered.

so I'm going to clear this also 

   Q. Find by what ratio point P (-3, 5) is dividing line segment with points (-7, 6) and (3, 4) ?

  
 Sol. let the ratio be k : 1

then again we'll apply the same section formula here 

for P( -3, 5)

-3 = [ k× 3 + 1× -7]  / ( k+ 1)

by cross multiplying

-3× (k + 1) = 3k -7

-3k -3 = 3k - 7

-6k = -4

k = -4/-6

k = 2/3

i.e.

 k/1 = 2/3

k : 1 =  2: 3 

here is one advantage we can have i.e. we don't bother to solve for x and y both value to find ratio it can be evaluated from either x or y.

Now I think this explanation of Section Formula fulfil your expectations regarding section formula,
If you are finding any doubts instead please comment under the section.