Question

What are the coordinates of the point on the directed line segment from (−9,−6) to (−2,1) that partitions the segment into a ratio of 3 to 4?

Answer 1

`\textit{internal division of a line segment using ratios} \\\\\\ A(-9,-6)\qquad B(-2,1)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:4} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{4}\implies \cfrac{A}{B} = \cfrac{3}{4}\implies 4A=3B\implies 4(-9,-6)=3(-2,1)`

`(\stackrel{x}{-36}~~,~~ \stackrel{y}{-24})=(\stackrel{x}{-6}~~,~~ \stackrel{y}{3}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-36 -6}}{3+4}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-24 +3}}{3+4} \right)} \\\\\\ C=\left( \cfrac{ -42 }{ 7 }~~,~~\cfrac{ -21}{ 7 } \right)\implies \boxed{C=(-6~~,~~-3)}`