Question

Find the point, M, that divides segment AB into a ratio of 4:7 if A is at (-33, 0) and B is at (0, 44)

Answer 1

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(-33,0)\qquad B(0,44)\qquad \qquad \stackrel{\textit{ratio from A to B}}{4:7} \\\\\\ \cfrac{A\underline{M}}{\underline{M} B} = \cfrac{4}{7}\implies \cfrac{A}{B} = \cfrac{4}{7}\implies 7A=4B\implies 7(-33,0)=4(0,44)\\\\[-0.35em] ~\dotfill\\\\ M=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill$

$\bf M=\left(\cfrac{(7\cdot -33)+(4\cdot 0)}{4+7}\quad ,\quad \cfrac{(7\cdot 0)+(4\cdot 44)}{4+7}\right) \\\\\\ M=\left(\cfrac{(-231)+(0)}{11}\quad ,\quad \cfrac{(0)+(176)}{11}\right)\\\\\\ M=\left( -\cfrac{231}{11}~,~\cfrac{176}{11} \right)\implies M=(-21,16)$