Question

What are the coordinates of the point on the directed line segment from (-6, 4) to (-2,-4) that partitions the segment into a ratio of 3 to 5?

Answer 1

let's say A(-6 , 4) and B(-2 , -4), so let's have point C partitions it from A to B in a 3:5 ratio, so

`\textit{internal division of a line segment using ratios} \\\\\\ A(-6,4)\qquad B(-2,-4)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:5} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{5}\implies \cfrac{A}{B} = \cfrac{3}{5}\implies 5A=3B\implies 5(-6,4)=3(-2,-4)`

`(\stackrel{x}{-30}~~,~~ \stackrel{y}{20})=(\stackrel{x}{-6}~~,~~ \stackrel{y}{-12}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-30 -6}}{3+5}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{20 -12}}{3+5} \right)} \\\\\\ C=\left( \cfrac{ -36 }{ 8 }~~,~~\cfrac{ 8}{ 8 } \right)\implies C=\left(-\cfrac{9}{2}~~,~~1 \right)\implies C=\left(-4(1)/(2)~~,~~1 \right)`