Coordinate Geometry

Section Formula

You decided to go go-karting from point A to B. Suddenly at a random point C your vehicle breaks down. How do you tell your friend who is waiting at the start line the details of your exact location? Well, just use the section formula. How? Find your answers by reading more in the below section.

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Section formula

Section Formula

When a point C divides a segment AB in the ratio m:n, we use the section formula to find the coordinates of that point. The section formula has 2 types. These types depend on the position of point C. It can be present between the 2 points or outside the segment.

The two types are:

  1. Internal Section Formula
  2. External Section Formula

Internal Section Formula

Also known as the Section Formula for Internal Division. When the line segment is divided internally in the ration m:n, we use this formula.  That is when the point C lies somewhere between the points A and B. Understand the concept of Coordinate here.

The Coordinates of point C will be,

{[(mx2+nx1)/(m+n)],[(my2+ny1)/(m+n)]}

Breaking it down, the x coordinate is (mx2+nx1)/(m+n) and the y coordinate is (my2+ny1)/(m+n)

Section Formula for External Division

When the point P lies on the external part of the line segment, we use the section formula for the external division for its coordinates.

A point on the external part of the segment means when you extend the segment than its actual length the point lies there. Just as you see in the diagram above. The section formula for external division is,

P={[(mx2-nx1)/(m-n)],[(my2-ny1)/(m-n)]}

Breaking it down, the x coordinate is (mx2-nx1)/(m-n) and the y coordinate is (my2-ny1)/(m-n)

Understand the concept of Distance Formula here.

Midpoint Formula

When we need to find the coordinates of a point that lies exactly at the center of any given segment we use the midpoint formula.

The midpoint formula is,

P={(x1+x2)/2,(y1+y2)/2}

Breaking it down, the x-coordinate is (x1+x2)/2 and the y-coordinate is (y1+y2)/2

Section Formulae at a Glance

For Internal Division P={[(mx2+nx1)/(m+n)],[(my2+ny1)/(m+n)]}
For External Division P={[(mx2-nx1)/(m-n)],[(my2-ny1)/(m-n)]}
Midpoint Formula P={(x1+x2)/2,(y1+y2)/2}

Solved Examples for You

Q: The point P divides the line segment AB joining points A(2,1) and B(-3,6) in the ratio 2:3. Does point P lie on the line x-5y+15=0? Justify.

Solution: Given that A(2,1)=(x1,y1), B(-3,6)=(x2,y2)

Point P divides the segment AB in the ratio 2:3, hence m=2, n=3

Since it isn’t mentioned in the question that the point divides the segment externally we use the section formula for internal division,

Formula: P={[(mx2+nx1)/(m+n)],[(my2+ny1)/(m+n)]}

Substituting all the known values,

={[(2(-3)+3(2))/(2+3)],[(2(6)+3(1))/(2+3)]}

=[(-6+6/5), (12+3/5)] =(0/5, 15/5)

Implies, P =(0,3)

To check if the point lies on the line x-5y+15=0 we substitute the coordinates of point P(0,3) in the equation.

LHS =x-5y+15 =0-5(3)+15 =0-15+15 =0 = RHS

Hence, the point P lies on the line x-5y+15=0

Q: Z (4, 5) and X(7, – 1) are two given points and the point Y divides the line-segment ZX externally in the ratio 4:3. Find the coordinates of Y.

Solution: Given that, Z(4,5)=(x1,y1), X(7,-1)=(x2,y2)

Point Y divides the segment ZX in the ratio 4:3, hence m=4, n=3

Since it is mentioned in the question that the point Y divides the segment externally we use the section formula for external division,

Formula: Y={[(mx2-nx1)/(m-n)],[(my2-ny1)/(m-n)]}

Substituting the known values,

={[(4(7)-3(4))/(4-3)],[(4(-1)-3(5)/(4-3)]}

={(28-12)/1,(-4-15)/1} ={16,-19}

The coordinates for the point Y are (16,-19)

Q: Find the midpoint of  segment AB where A(2,3) and B(6,7).

Solution: Given that, A(2,3)=(x1,y1), B(6,7)=(x2,y2)

Formula: P={(x1+x2)/2,(y1+y2)/2}

Substituting the known values,

P={(2+6)/2,(3+7)/2} ={8/2,10/2} =(4,5)

The Midpoint of the seg AB is (4,5).

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Derivation of section formula in all four quadrants

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