[tex]\bf \textit{internal division of a line segment using ratios} \\\\\\ A(1,2)\qquad B(8,16)\qquad \qquad \stackrel{\textit{ratio from A to B}}{5:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{5}{2}\implies \cfrac{A}{B} = \cfrac{5}{2}\implies 2A=5B\implies 2(1,2)=5(8,16)\\\\[-0.35em] ~\dotfill\\\\ M=\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 M=\left(\cfrac{(2\cdot 1)+(5\cdot 8)}{5+2}\quad ,\quad \cfrac{(2\cdot 2)+(5\cdot 16)}{5+2}\right) \\\\\\ M=\left( \cfrac{2+40}{7}~~,~~\cfrac{4+80}{7} \right)\implies M=\left( \cfrac{42}{7}~,~\cfrac{84}{7} \right)\implies M=(6~,~12)[/tex]
