# Internal Division of Line Segments

Internal Division an interval in a given ratio is that if the interval $(x_1,y_1)$ and $(x_2,y_2)$ is divided in the ratio $m:n$ then the coordinates are;
$$\Big(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\Big)$$

## Example 1: Basics

If $A$ and $B$ are the points $(-4,3)$ and $(2,-1)$ respectively, find the coordinates of $P$ such that $AP:PB=3:1$.

\begin{aligned} \displaystyle P(x,y) &= \Big(\frac{3 \times 2 + 1 \times -4}{3+1},\frac{3 \times -1 + 1 \times 3}{3+1}\Big) \\ &= \Big(\frac{1}{2},0\Big) \\ \end{aligned} \\

## Example 2: Negative Ratio of Internal Division of Line Segments

If $A$ and $B$ are the points $(-4,3)$ and $(2,-1)$ respectively, find the coordinates of $P$ such that $AP:PB=-4:5$.

\begin{aligned} \displaystyle P(x,y) &= \Big(\frac{-4 \times 2 + 5 \times -4}{-4+5},\frac{-4 \times -1 + 5 \times 3}{-4+5}\Big) \\ &= (-28,19) \\ \end{aligned} \\

## Example 3: Three Equal Parts using Internal Division of Line Segments

Divide the interval between $(-1,1)$ and $(5,10)$ into three equal parts.

\begin{aligned} \displaystyle (a,b) &= \Big(\frac{1 \times 5 + 2 \times -1}{1+2},\frac{1 \times 10 + 2 \times 1}{1+2}\Big) \\ &= (1,4) \\ (c,d) &= \Big(\frac{2 \times 5 + 1 \times -1}{2+1},\frac{2 \times 10 + 1 \times 1}{2+1}\Big) \\ &= (3,7) \\ \end{aligned} \\

## Example 4: Finding the Ratio of Internal Division of Line Segments

If the point $(-3,8)$ divides the interval between $(6,-4)$ and $(0,4)$ internally in the ratio $k:1$, find the value of $k$.

\begin{aligned} \displaystyle (-3,8) &= \Big(\frac{k \times 0 + 1 \times 6}{k+1},\frac{k \times 4 + 1 \times -4}{k+1}\Big) \\ &= \Big(\frac{6}{k+1},\frac{4k-4}{k+1}\Big) \\ \frac{6}{k+1} &= -3 \\ k+1 &= 6 \div -3 \\ k+1 &= -2 \\ \therefore k &= -3 \\ \end{aligned} \\