Coordinate Geometry · Topic 2 of 6

✂️ Section Formula & Centroid

2 exam question types, fully solved

The section formula finds a point that splits a segment in a given ratio. The midpoint is just the special case 1:1, and a triangle's centroid is the average of its three corners.

Cross-multiply the ratio

• Internal division of and : the point is . Note the cross pairing — multiplies the far point's coordinate.

• Midpoint is the ratio , which collapses to the average.

• Centroid of a triangle: — it sits on each median, dividing it from the vertex.

📐Coordinate Plane ExplorerLive calculations as you move points

A (x₁, y₁)

x₁

y₁

B (x₂, y₂)

x₂

y₂

Ratio m : n

P divides AB internally in 1 : 2((m·x₂+n·x₁)/(m+n), (m·y₂+n·y₁)/(m+n)) = ((1·4+2·1)/3, (1·5+2·2)/3)P = (2, 3)

Check the cross pairingm = 1 multiplies the FAR point B(4, 5); n = 2 multiplies the NEAR point A(1, 2)P is 1 part from A, 2 parts from B

Slide m : n and watch P move along AB — bigger m pulls P towards B, bigger n keeps it near A. At 1 : 1 you get the midpoint.

The point dividing (0, 0) and (10, 0) in the ratio 2:3 has x-coordinate .

Q1The point dividing (0, 0) and (6, 9) internally in the ratio 1:2 is:

Q2The centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9) is:

Q3A point divides (2, 3) and (8, 11) in the ratio 1:1. It is the:

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Real exam questions — Section Formula & Centroid

2 question types · 8 solved examples from real SSC papers

Section-formula and centroid questions are pure plug-and-go: the section formula is a weighted average of two points, and the centroid is the plain average of three. Master which ratio number pairs with which point, always divide the centroid by 3, and most of these fall in under a minute.

How to solve this type

A point

P

dividing the join of

A(x_1,y_1)

and

B(x_2,y_2)

internally in the ratio

m:n

has coordinates

\left(\dfrac{mx_2+nx_1}{m+n},\,\dfrac{my_2+ny_1}{m+n}\right)

. The golden rule: the FIRST ratio number

m

pairs with the FAR point

B(x_2,y_2)

, and

n

pairs with the near point

A(x_1,y_1)

— internal division always uses PLUS signs. To find the ratio instead, use

m:n=(x-x_1):(x_2-x)

. Two time-savers: if the point is the average of the ends, the ratio is

1:1

(midpoint); and you usually only need ONE coordinate to pick the option.

Points $A(2,4)$ and $B(8,10)$ are given. Find the point dividing $AB$ internally in the ratio $2:1$ .

(7,9)

(6,8)

(5,7)

(4,6)

Find the point $P$ dividing the segment joining $A(-1,3)$ and $B(4,-2)$ internally in the ratio $2:3$ .

(1,0)

(1,1)

(0,1)

(2,0)

Point $P(3,y)$ divides the join of $A(1,2)$ and $B(7,8)$ in the ratio $1:2$ . Find $y$ .

6

4

3

5

In what ratio does the point $(3,4)$ divide the segment joining $(1,2)$ and $(7,8)$ ?

2:3

1:2

2:1

1:3