Question

What are the coordinates of the point on the directed line segments from (-2,-5) to (4,1) that partitions the segment into a ratio of 1 to 2?

Answer 1

let's say that A(-2,-5) to B(4,1) gets partitioned by point C into a ratio of 1 to 2

`\textit{internal division of a line segment using ratios} \\\\\\ A(-2,-5)\qquad B(4,1)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{2}\implies \cfrac{A}{B} = \cfrac{1}{2}\implies 2A=1B\implies 2(-2,-5)=1(4,1)`

`(\stackrel{x}{-4}~~,~~ \stackrel{y}{-10})=(\stackrel{x}{4}~~,~~ \stackrel{y}{1}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-4 +4}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-10 +1}}{1+2} \right)} \\\\\\ C=\left( \cfrac{ 0 }{ 3 }~~,~~\cfrac{ -9}{ 3 } \right)\implies C=(0~~,~-3)`