ICSE Class 10 Maths: Section Formula for Dividing a Line Segment

Why the Section Formula is a Handy Tool in Coordinate Geometry

Ever wondered how to find a point that splits a line exactly the way you want? That’s what the section formula does – it’s like a GPS for points on a straight road.

In simple words: If you know the ends of a line and how many parts you want to split it into, the section formula tells you the exact coordinates of the dividing point.

What is the Section Formula?

The section formula gives the coordinates of a point that divides a line segment joining two points A(x₁, y₁) and B(x₂, y₂) in a given ratio m:n. The ratio tells you how many “shares” each end gets. If the point is between A and B, we call it internal division. If it lies outside the segment, it’s an external division.

How the Formula Looks

• Internal division: \(P\big(\frac{mx₂+nx₁}{m+n}, \frac{my₂+ny₁}{m+n}\big)\)
• External division: \(P\big(\frac{mx₂-nx₁}{m-n}, \frac{my₂-ny₁}{m-n}\big)\)

Think of the line as a water pipe. The ratio m:n is like saying “let 3 parts of the water flow from the left end and 2 parts from the right end.” The formula tells you where the mixing point is.

How to Use It: Step‑by‑Step

Worked Example 1 – Internal Division

Find the point that divides the line joining A(2, 3) and B(8, 9) in the ratio 3:2.

1. Write down the coordinates: \(x₁=2, y₁=3, x₂=8, y₂=9\).
2. Use the internal formula: \(x = \frac{m x₂ + n x₁}{m+n}\), \(y = \frac{m y₂ + n y₁}{m+n}\).
3. Plug in \(m=3, n=2\):
   • \(x = \frac{3\times8 + 2\times2}{3+2} = \frac{24+4}{5} = \frac{28}{5} = 5.6\)
   • \(y = \frac{3\times9 + 2\times3}{5} = \frac{27+6}{5} = \frac{33}{5} = 6.6\)
4. So the required point is \(P(5.6, 6.6)\).

Worked Example 2 – External Division

Find the point that divides the line joining C(4, 1) and D(10, 7) externally in the ratio 1:2.

1. Coordinates: \(x₁=4, y₁=1, x₂=10, y₂=7\).
2. External formula: \(x = \frac{m x₂ - n x₁}{m-n}\), \(y = \frac{m y₂ - n y₁}{m-n}\).
3. Insert \(m=1, n=2\):
   • \(x = \frac{1\times10 - 2\times4}{1-2} = \frac{10-8}{-1} = \frac{2}{-1} = -2\)
   • \(y = \frac{1\times7 - 2\times1}{-1} = \frac{7-2}{-1} = \frac{5}{-1} = -5\)
4. Hence the external point is \(P(-2, -5)\), which lies on the line extension beyond C.

Quick Reference Table

Common Mistakes to Avoid

• Mixing up \(m\) and \(n\). The first number always multiplies the far end’s coordinates.
• Using the internal formula when the point is outside the segment.
• Forgetting to simplify the denominator; a zero denominator means the ratio is impossible.
• Skipping the sign check in external division – the denominator becomes \(m-n\), not \(m+n\).

📝 Likely Exam Questions

1. Question: Find the coordinates of the point that divides the segment joining \((1, 4)\) and \((7, 10)\) in the ratio 5:3 internally.
   Answer: \(x = \frac{5\times7 + 3\times1}{8}=\frac{35+3}{8}=\frac{38}{8}=4.75\), \(y = \frac{5\times10 + 3\times4}{8}=\frac{50+12}{8}=\frac{62}{8}=7.75\). Point = \((4.75, 7.75)\).
2. Question: A line joins \((‑2, 5)\) and \((6, ‑3)\). Determine the point that divides it externally in the ratio 2:1.
   Answer: \(x = \frac{2\times6 - 1\times(‑2)}{2‑1}=\frac{12+2}{1}=14\), \(y = \frac{2\times(‑3) - 1\times5}{1}=\frac{‑6‑5}{1}=‑11\). Point = \((14, ‑11)\).
3. Question: If a point \(P\) divides the segment joining \((3, 2)\) and \((9, 8)\) in the ratio 1:1, what are the coordinates of \(P\)?
   Answer: Midpoint (special case of section formula): \(x = \frac{3+9}{2}=6\), \(y = \frac{2+8}{2}=5\). So \(P(6, 5)\).
4. Question: Explain why the denominator becomes \(m‑n\) for external division.
   Answer: In external division the point lies outside, so the directed distances subtract rather than add, giving a net “effective” ratio of \(m‑n\).
5. Question: A line segment has endpoints \((0, 0)\) and \((12, 0)\). Find the point that is three‑quarters of the way from the first endpoint.
   Answer: Ratio = 3:1 (since 3 parts from the start, 1 part from the end). Using internal formula: \(x = \frac{3\times12 + 1\times0}{4}=9\), \(y = 0\). Point = \((9, 0)\).