7.1	Division of a Line Segment

A point \(P(x,y)\) that divides a line segment joining  \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio of \(m:n\) is given by

\(\boxed{P=\left(\dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+my_2}{m+n} \right)}\)

For the case \(m=n\), the point is a midpoint and is given by

\(\boxed{M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2} \right)}\)

Example: Find the coordinates of point \(P\) which divides the straight line \(AB\) in the ratio \(AP:PB\).

\(\begin{aligned} &A(-3,2),B(7,17);\\\\&AP:PB=2:3 \end{aligned}\)

\(\begin{aligned} P&=\left(\dfrac{2(7)+3(-3)}{2+3},\dfrac{2(17)+3(2)}{2+3} \right) \\\\ &= (1,8). \end{aligned}\)