[tex]\textit{internal division of a line segment using ratios} \\\\\\ A(-7,2)\qquad B(1,-6)\qquad \qquad \stackrel{\textit{ratio from A to B}}{5:3} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{5}{3}\implies \cfrac{A}{B} = \cfrac{5}{3}\implies 3A=5B\implies 3(-7,2)=5(1,-6)[/tex]
[tex](\stackrel{x}{-21}~~,~~ \stackrel{y}{6})=(\stackrel{x}{5}~~,~~ \stackrel{y}{-30})\implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-21+5}}{5+3}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{6-30}}{5+3} \right)} \\\\\\ P=\left( \cfrac{-16}{8}~~,~~\cfrac{-24}{8} \right)\implies P=(-2~~,~~-3)[/tex]
