# Internal Division Section formula

## Formula

$P(x, y)$ $\,=\,$ $\bigg(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\bigg)$

A formula that helps us to find the coordinates of a point which internally divides the line joining two points in a specific ratio is called the internal division section formula.

### Introduction

A point on a line segment divides the line internally in a specific ratio. The coordinates of the point on the line segment can be evaluated from the coordinates of the both endpoints of that line and the internal division ratio.

Let $x_1$ and $y_1$ be coordinates of one endpoint of the line segment and $x_2$ and $y_2$ be coordinates of another endpoint of the line segment in a two dimensional cartesian coordinate system.

If $x$ and $y$ are the coordinates of the point which internally divides the line in a ratio of $m : n$, then the coordinates of the point can be calculated from the below formula.

$P(x, y)$ $\,=\,$ $\bigg(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\bigg)$

This formula is called the internal division section formula.

#### Proof

Learn how to derive the internal division section formula in algebraic form geometrically.

#### Problems

The list of geometry problems with solutions to learn how to find the coordinates of a point which internally divides a line in a given ratio by using the internal division section formula.

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