1. Introduction
The section formula helps us find the coordinates of a point that divides a line segment joining two points in a given ratio. This point can divide the segment internally (between the two points) or externally (outside the segment).
If you know two endpoints and the ratio in which a point divides the segment, this formula gives the exact coordinates of that dividing point.
2. Understanding Internal Division
Internal division means the point lies between the two given points A and B.
If A is \((x_1, y_1)\) and B is \((x_2, y_2)\), and point P divides AB in ratio m : n, then:
2.1. Internal Division Formula
The coordinates of P are:
\(P(x, y) = \left( \dfrac{mx_2 + nx_1}{m+n}, \dfrac{my_2 + ny_1}{m+n} \right) \)
This gives a weighted average of the coordinates.
3. Understanding External Division
External division means the point lies outside the line segment AB.
The same idea applies, but the formula changes slightly because the point is not between the endpoints.
3.1. External Division Formula
If P divides AB externally in the ratio m : n, then:
\(P(x, y) = \left( \dfrac{mx_2 - nx_1}{m-n}, \dfrac{my_2 - ny_1}{m-n} \right) \)
Use this only when m ≠ n.
4. Why the Formula Works (Simple Idea)
The formula is based on the idea of a weighted average. If the ratio is m : n:
- The point is closer to the endpoint with the smaller weight.
- The coordinates shift toward the point with higher influence.
It is like balancing a see-saw — the ratios decide where the middle settles.
5. Worked Examples
Let's understand the formula with clear examples.
5.1. Example 1: Internal Division
A(2, 3), B(8, 9). Find the point dividing AB in ratio 1 : 2.
Using formula:
\(P(x, y) = \left( \dfrac{1·8 + 2·2}{1+2}, \dfrac{1·9 + 2·3}{1+2} \right)\)
\( = \left( \dfrac{8 + 4}{3}, \dfrac{9 + 6}{3} \right) = (4, 5)\)
5.2. Example 2: Internal Division (Different Ratio)
A(-4, 6), B(10, -2). Find the point dividing AB in ratio 3 : 5.
\(P = \left( \dfrac{3·10 + 5(-4)}{8}, \dfrac{3(-2) + 5·6}{8} \right)\)
\( = \left( \dfrac{30 - 20}{8}, \dfrac{-6 + 30}{8} \right) = (1.25, 3)\)
5.3. Example 3: External Division
A(1, 2), B(5, 6). Find the point dividing AB externally in ratio 2 : 1.
\(P = \left( \dfrac{2·5 - 1·1}{2-1}, \dfrac{2·6 - 1·2}{2-1} \right)\)
\( = (9, 10)\)
6. Common Mistakes
- Using the internal formula for external division.
- Forgetting that denominator is (m + n) for internal and (m − n) for external.
- Mixing up x- and y-coordinates.
- Incorrect sign handling when x1 or x2 are negative.
- Using external formula when m = n (not allowed).
7. Quick Practice
Try these:
- Find the point dividing (3, 1) and (9, 7) in ratio 2 : 3 internally.
- Find the point dividing (-4, -3) and (6, 5) in ratio 5 : 1 internally.
- Using external division, divide points (2, 4) and (10, 12) in ratio 3 : 2.
- Find the coordinates of a point that divides AB internally in ratio 1 : 1 (what do you get?).
8. Summary
- Section formula gives coordinates of a point dividing a line in a given ratio.
- Internal division: \(P = \left( \dfrac{mx_2 + nx_1}{m+n}, \dfrac{my_2 + ny_1}{m+n} \right)\)
- External division: \(P = \left( \dfrac{mx_2 - nx_1}{m-n}, \dfrac{my_2 - ny_1}{m-n} \right)\)
- The concept is based on weighted averages of coordinates.