Question

A directed line segment AB with A(3,2) and B (6,11) is partitioned by point C such that AC and CB form a 2:1 ratio. Find C. Write your answer as a coordinate point. Do not include spaces in your answer.

Answer 1

$\bf \textit{internal division of a line segment using ratios} \\\\\\ A(3,2)\qquad B(6,11)\qquad \qquad \stackrel{\textit{ratio from A to B}}{2:1} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{2}{1}\implies \cfrac{A}{B} = \cfrac{2}{1}\implies 1A=2B\implies 1(3,2)=2(6,11)\\\\[-0.35em] ~\dotfill\\\\ C=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill$

$\bf C=\left(\cfrac{(1\cdot 3)+(2\cdot 6)}{2+1}\quad ,\quad \cfrac{(1\cdot 2)+(2\cdot 11)}{2+1}\right) \\\\\\ C=\left( \cfrac{3+12}{3}~~,~~\cfrac{2+22}{3} \right)\implies C=\left( \cfrac{15}{3}~~,~~\cfrac{24}{3} \right)\implies C=(5~~,~~8)$