Question

Point C with coordinates (-3.6, -3.4) divides in the ratio 2 : 3. If A = (-2, -1), what are the coordinates of point B

Answer 1

$\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ A(-2,-1)\qquad B(x,y)\qquad \qquad 2:3 \\\\\\ \cfrac{AC}{CB} = \cfrac{2}{3}\implies \cfrac{A}{B} = \cfrac{2}{3}\implies 3A=2B\implies 3(-2,-1)=2(x,y)\\\\ -------------------------------\\\\ { C=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\ -------------------------------$

$\bf C=\left(\cfrac{(3\cdot -2)+(2\cdot x)}{2+3}\quad ,\quad \cfrac{(3\cdot -1)+(2\cdot y)}{2+3}\right)~~=~~(-3.6~~,~~-3.4) \\\\\\ C=\left(\cfrac{-6+2x}{5}~~,~~\cfrac{-3+2y}{5} \right)~~~=~~~(-3.6~~,~~-3.4)\\\\ -------------------------------\\\\ \cfrac{-6+2x}{5}=-3.6\implies -6+2x=-18\implies 2x=-12 \\\\\\ x=\cfrac{-12}{2}\implies \boxed{x=-6}\\\\ -------------------------------\\\\ \cfrac{-3+2y}{5} =-3.4\implies -3+2y=-17 \\\\\\ 2y=-14\implies y=\cfrac{-14}{2}\implies \boxed{y=-7}$