Question

What are the coordinates of the point on the directed line segment from (−7,−5) to (8,−10) that partitions the segment into a ratio of 3 to 2?

Answer 1

let's say P(−7,−5) and Q(8,−10) gets partitioned on a 3 : 2 ratio from P to Q by point R, so

`\textit{internal division of a line segment using ratios} \\\\\\ P(-7,-5)\qquad Q(8,-10)\qquad \qquad \stackrel{\textit{ratio from P to Q}}{3:2} \\\\\\ \cfrac{P\underline{R}}{\underline{R} Q} = \cfrac{3}{2}\implies \cfrac{P}{Q} = \cfrac{3}{2}\implies 2P=3Q\implies 2(-7,-5)=3(8,-10)`

`(\stackrel{x}{-14}~~,~~ \stackrel{y}{-10})=(\stackrel{x}{24}~~,~~ \stackrel{y}{-30}) \implies R=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-14 +24}}{3+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-10 -30}}{3+2} \right)} \\\\\\ R=\left( \cfrac{ 10 }{ 5 }~~,~~\cfrac{ -40}{ 5 } \right)\implies R=(2~~,~-8)`