[tex]\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ P(2,6)\qquad Q(6,14)\qquad \qquad \stackrel{\textit{ratio from P to Q}}{4:5} \\\\\\ \cfrac{P\underline{N}}{\underline{N} Q} = \cfrac{4}{5}\implies \cfrac{P}{Q} = \cfrac{4}{5}\implies 5P=4Q\implies 5(2,6)=4(6,14)\\\\[-0.35em] ~\dotfill\\\\ N=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf N=\left(\cfrac{(5\cdot 2)+(4\cdot 6)}{4+5}\quad ,\quad \cfrac{(5\cdot 6)+(4\cdot 14)}{4+5}\right)\implies N=\left(\cfrac{34}{9}~,~\cfrac{86}{9} \right) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill N=\left( 3\frac{7}{9}~~,~~9\frac{5}{9} \right)~\hfill[/tex]
