Section Formula (Internal)

Statement

If a point \(P\) divides the line segment between \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) internally, then its coordinates are:

\[ P = \left( \frac{mx_2 + nx_1}{m+n}, \; \frac{my_2 + ny_1}{m+n} \right) \]

Why it’s true

• The formula is a weighted average of the coordinates.
• If the ratio is 1:1, it gives the midpoint formula.
• The weights \(m\) and \(n\) determine how close \(P\) is to each endpoint.
• Larger \(m\) pulls \(P\) closer to \(B(x_2,y_2)\), larger \(n\) pulls it toward \(A(x_1,y_1)\).

Recipe (how to use it)

1. Identify the endpoints \(A(x_1,y_1)\) and \(B(x_2,y_2)\).
2. Write down the ratio \(m:n\).
3. Substitute into the formula for both \(x\) and \(y\) coordinates.
4. Simplify fractions to get the coordinates of \(P\).

Spotting it

Look for problems saying a point divides a line in a certain ratio internally, e.g., “Find the point that divides AB in the ratio 2:3.”

Common pairings

• Midpoints (special case of ratio 1:1).
• Geometry of triangles (e.g. centroid divides medians in 2:1).
• Coordinate geometry and vector problems.

Mini examples

1. Find point dividing A(2,4) and B(10,8) in ratio 1:1 → \((6,6)\).
2. Find point dividing A(1,2) and B(7,8) in ratio 2:3 → \((5,6)\).
3. Find point dividing A(-3,5) and B(9,1) in ratio 3:1 → \((6,2)\).

Pitfalls

• Mixing up order of \(m\) and \(n\).
• Forgetting it’s internal division (both weights positive).
• Not simplifying coordinates fully.

Exam strategy

• Write formula first before substituting values.
• Double-check placement of ratio values with endpoints.
• Use midpoint formula as a quick check if ratio=1:1.

Summary

The internal section formula gives the coordinates of a point dividing a line in a chosen ratio, using weighted averages of the endpoints.