Question

For the directed line segment whose endpoints are (4,3) and (-8,-5), find the coordinates of the point that partitions the segment into a ratio of 3 to 1

Answer 1

$\textit{internal division of a line segment using ratios} \\\\\\ A(4,3)\qquad B(-8,-5)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:1} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{1}\implies \cfrac{A}{B} = \cfrac{3}{1}\implies 1A=3B\implies 1(4,3)=3(-8,-5)$

$(\stackrel{x}{4}~~,~~ \stackrel{y}{3})=(\stackrel{x}{-24}~~,~~ \stackrel{y}{-15})\implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{4-24}}{3+1}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{3-15}}{3+1} \right)} \\\\\\ C=\left(\cfrac{-20}{4}~~,~~\cfrac{-12}{4} \right)\implies C=(-5~~,~~-3)$