Question

Find the point that is two-thirds of the way from (3,4) to (9,-2)

Answer 1

Keeping in mind that, the segment is split in three thirds, and the point is 2/3 of the way, so is closer to (9, -2), with a 2:1 ratio, check the picture below.

$\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ A(3,4)\qquad B(9,-2)\qquad \qquad 2:1 \\\\\\ \cfrac{AC}{CB} = \cfrac{2}{1}\implies \cfrac{A}{B} = \cfrac{2}{1}\implies 1A=2B\implies 1(3,4)=2(9,-2)\\\\ -------------------------------\\\\ { C=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\ -------------------------------$

$\bf C=\left(\cfrac{(1\cdot 3)+(2\cdot 9)}{2+1}\quad ,\quad \cfrac{(1\cdot 4)+(2\cdot -2)}{2+1}\right) \\\\\\ C=\left( \cfrac{3+18}{3}~~,~~\cfrac{4-4}{3} \right)\implies C=\left( 7~~,~~0 \right)$