Section Formula in Coordinate Geometry

The Section Formula is a fundamental concept in coordinate geometry that allows us to find the coordinates of a point that divides a line segment into a particular ratio. This formula is applicable in both two-dimensional and three-dimensional geometry. It is particularly useful in problems where we need to find the midpoint of a line segment, the centroid of a triangle, or any point that divides a line segment in a given ratio.

The Section Formula in 2D

In two-dimensional geometry, the Section Formula is used to find the coordinates of a point P(x, y) that divides the line segment joining two points A(x_1, y_1) and B(x_2, y_2) in the ratio m:n.

Internal Division

When a point divides a line segment internally, the coordinates of the point P are given by:

$$ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n} $$

External Division

When a point divides a line segment externally, the coordinates of the point P are given by:

$$ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n} $$

Midpoint Formula

A special case of the Section Formula is the Midpoint Formula, where m = n = 1. The coordinates of the midpoint M of the line segment AB are:

$$ x_M = \frac{x_1 + x_2}{2}, \quad y_M = \frac{y_1 + y_2}{2} $$

The Section Formula in 3D

In three-dimensional geometry, the Section Formula is extended to include the z-coordinate. If A(x_1, y_1, z_1) and B(x_2, y_2, z_2) are two points in space, and P(x, y, z) divides AB in the ratio m:n, then:

Internal Division in 3D

$$ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}, \quad z = \frac{m z_2 + n z_1}{m + n} $$

External Division in 3D

$$ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n}, \quad z = \frac{m z_2 - n z_1}{m - n} $$

Differences and Important Points

Here is a table summarizing the differences and important points of the Section Formula:

Aspect	Internal Division	External Division
Ratio	Positive m:n	Positive m:n
Formula (2D)	(mx_2 + nx_1)/(m + n)	(mx_2 - nx_1)/(m - n)
Formula (3D)	Same as 2D with z added	Same as 2D with z added
Midpoint (Special Case)	m = n = 1	Not applicable
Application	Finding points on a line	Finding points out of line

Examples

Example 1: Internal Division in 2D

Find the coordinates of the point which divides the line segment joining A(2, 3) and B(5, 7) in the ratio 2:3 internally.

Solution:

Using the internal division formula:

$$ x = \frac{2 \cdot 5 + 3 \cdot 2}{2 + 3} = \frac{10 + 6}{5} = \frac{16}{5} = 3.2 $$

$$ y = \frac{2 \cdot 7 + 3 \cdot 3}{2 + 3} = \frac{14 + 9}{5} = \frac{23}{5} = 4.6 $$

So, the coordinates of the point are (3.2, 4.6).

Example 2: External Division in 3D

Find the coordinates of the point which divides the line segment joining A(1, 2, 3) and B(4, 6, 8) in the ratio 1:2 externally.

Solution:

Using the external division formula:

$$ x = \frac{1 \cdot 4 - 2 \cdot 1}{1 - 2} = \frac{4 - 2}{-1} = -2 $$

$$ y = \frac{1 \cdot 6 - 2 \cdot 2}{1 - 2} = \frac{6 - 4}{-1} = -2 $$

$$ z = \frac{1 \cdot 8 - 2 \cdot 3}{1 - 2} = \frac{8 - 6}{-1} = -2 $$

So, the coordinates of the point are (-2, -2, -2).

Understanding the Section Formula is crucial for solving problems in coordinate geometry. It is a versatile tool that can be applied to various scenarios, including finding the centroid of a triangle, the point of trisection, and more.