Section Formula & Mid-Point Formula | Coordinate Geometry | Class 10

01 · SECTION FORMULA

Internal Division in Ratio m₁ : m₂

The coordinates of point P(x, y) which divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are:

P(x, y) = ( (m₁x₂ + m₂x₁) / (m₁ + m₂), (m₁y₂ + m₂y₁) / (m₁ + m₂) )

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The k : 1 Ratio Trick

When finding the ratio in which point P divides line segment AB, assume the ratio to be k : 1. The coordinates become:
P(x, y) = ( (kx₂ + x₁) / (k + 1), (ky₂ + y₁) / (k + 1) )

02 · MID-POINT FORMULA

Mid-Point of a Line Segment

If P is the mid-point of line segment AB, the ratio is 1 : 1 (m₁ = 1, m₂ = 1). The formula simplifies to:

Mid-Point M = ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 )

03 · SOLVED NCERT EXAMPLES

Step-by-Step Problem Solving

EXAMPLE 1

Find coordinates of point P dividing A(4, -3) and B(8, 5) in ratio 3 : 1

Given: x₁ = 4, y₁ = -3, x₂ = 8, y₂ = 5, m₁ = 3, m₂ = 1.

x = (3 × 8 + 1 × 4) / (3 + 1) = (24 + 4) / 4 = 28/4 = 7
y = (3 × 5 + 1 × (-3)) / (3 + 1) = (15 - 3) / 4 = 12/4 = 3

Coordinates of P are (7, 3).

EXAMPLE 2

Find the ratio in which Y-axis divides A(-4, 2) and B(3, -5)

Since the point lies on the Y-axis, its x-coordinate is 0. Let ratio = k : 1.

x = (k(3) + 1(-4)) / (k + 1) = 0
3k - 4 = 0 ⟹ k = 4/3

The Y-axis divides line segment AB in the ratio 4 : 3.

04 · QUICK SUMMARY

Key Takeaways

Section formula coordinates: ( (m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂) ).

Mid-point formula: ( (x₁ + x₂)/2, (y₁ + y₂)/2 ).

Point on X-axis has coordinates (x, 0); point on Y-axis has coordinates (0, y).