Section Formula (Internal Division)

The section formula is a mathematical expression used to find the coordinates of a point that divides a line segment internally in a given ratio. This formula is particularly useful in coordinate geometry and vector algebra.

Understanding the Section Formula

Consider a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$. Let $P(x, y)$ be a point on the line segment AB that divides it in the ratio $m:n$ internally. The coordinates of point P can be determined using the section formula.

The Formula

The section formula for internal division states that the coordinates of the point P are given by:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} $$

Table of Differences and Important Points

Aspect	Description
Division Type	Internal division of a line segment
Ratio	The point divides the segment in a specific ratio $m:n$
Formula	$P(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$
Special Cases	When $m = n$, the point P is the midpoint of AB
Applicability	Applicable in both 2D and 3D coordinate geometry

Examples

Example 1: Midpoint

Find the midpoint of the line segment with endpoints $A(2, 3)$ and $B(4, 7)$.

Solution:

For the midpoint, the ratio $m:n$ is $1:1$. Applying the section formula:

$$ x = \frac{1 \cdot 4 + 1 \cdot 2}{1 + 1} = \frac{6}{2} = 3 $$

$$ y = \frac{1 \cdot 7 + 1 \cdot 3}{1 + 1} = \frac{10}{2} = 5 $$

Therefore, the midpoint is $(3, 5)$.

Example 2: Internal Division in Ratio 2:3

Find the coordinates of the point which divides the line segment joining $A(-2, 5)$ and $B(4, -3)$ in the ratio 2:3 internally.

Solution:

Using the section formula with $m = 2$ and $n = 3$:

$$ x = \frac{2 \cdot 4 + 3 \cdot (-2)}{2 + 3} = \frac{8 - 6}{5} = \frac{2}{5} $$

$$ y = \frac{2 \cdot (-3) + 3 \cdot 5}{2 + 3} = \frac{-6 + 15}{5} = \frac{9}{5} $$

The coordinates of the point are $\left(\frac{2}{5}, \frac{9}{5}\right)$.

Example 3: 3D Coordinates

Find the point which divides the line segment joining $A(1, 2, 3)$ and $B(4, 5, 6)$ in the ratio 1:2 internally.

Solution:

For 3D coordinates, the section formula is extended to include the z-coordinate:

$$ x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = 2 $$

$$ y = \frac{1 \cdot 5 + 2 \cdot 2}{1 + 2} = \frac{5 + 4}{3} = 3 $$

$$ z = \frac{1 \cdot 6 + 2 \cdot 3}{1 + 2} = \frac{6 + 6}{3} = 4 $$

The point is $(2, 3, 4)$.

Conclusion

The section formula for internal division is a powerful tool in coordinate geometry and vector algebra. It allows us to find the coordinates of a point that divides a line segment in a given ratio. By understanding and applying this formula, one can solve various problems related to points on a line segment.