[tex]\textit{internal division of a line segment using ratios} \\\\\\ A(6,5)\qquad B(6,15)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:2} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{3}{2}\implies \cfrac{A}{B} = \cfrac{3}{2}\implies 2A=3B\implies 2(6,5)=3(6,15)[/tex]
[tex](\stackrel{x}{12}~~,~~ \stackrel{y}{10})=(\stackrel{x}{18}~~,~~ \stackrel{y}{45})\implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{12 +18}}{3+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{10 +45}}{3+2} \right)} \\\\\\ P=\left( \cfrac{30}{5}~~,~~\cfrac{55}{5} \right)\implies P=(6~~,~~11)[/tex]
