Question

Points A (-10, -6) and B (6, 2) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 3/4ths the length of AB?

Answer 1

we know that AC is 3/4 of the way from AB, meaning, if we cut AB in 4 equal pieces, AC takes 3 of those, and CB takes 1, meaning, AC and CB are on a 3:1 ratio.

`\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(-10,-6)\qquad B(6,2)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:1} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{1}\implies \cfrac{A}{B} = \cfrac{3}{1}\implies 1A=3B\implies 1(-10,-6)=3(6,2)\\\\[-0.35em] ~\dotfill\\\\ C=\left(\frac{\textit{sum of`

`\bf C=\left(\cfrac{(1\cdot -10)+(3\cdot 6)}{3+1}\quad ,\quad \cfrac{(1\cdot -6)+(3\cdot 2)}{3+1}\right) \\\\\\ C=\left( \cfrac{-10+18}{4}~~,~~\cfrac{-6+6}{4} \right)\implies C=\left( \cfrac{8}{4}~,~\cfrac{0}{4} \right)\implies C=(2,0)`