Question

The endpoints of MP are M(1,4) amd P(16,14). If A partitions MP in a ratio of MA:AP = 2:1, which of the following represent the coordinates of point A?

Answer 1

So, we know the segment MP gets partitioned by the point A to MA with a ratio of 2 and AP with a ratio of 1, on a 2:1 ratio from M to P, therefore then,

$\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ M(1,4)\qquad P(16,14)\qquad \qquad 2:1 \\\\\\ \cfrac{MA}{AP} = \cfrac{2}{1}\implies \cfrac{M}{P} = \cfrac{2}{1}\implies 1M=2P\implies 1(1,4)=2(16,14)\\\\ -------------------------------\\\\ { A=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}$

$\bf -------------------------------\\\\ A=\left(\cfrac{(1\cdot 1)+(2\cdot 16)}{2+1}\quad ,\quad \cfrac{(1\cdot 4)+(2\cdot 14)}{2+1}\right) \\\\\\ A=\left( \cfrac{1+32}{3}~~,~~\cfrac{4+28}{3} \right)\implies A=\left( \cfrac{33}{3}~~,~~\cfrac{32}{3} \right) \\\\\\ A=\left(11~~,10\frac{2}{3} \right)$