Question

What are the coordinates of the point on the directed line segment from (−3,8) to (6, -4) that partitions the segment into a ratio of 1 to 5?

Answer 1

let's say the segment is A(-3,8) and B(6,-4) and point C splits it on a 1:5 ratio from A to B, let's check for C coordinates.

`\textit{internal division of a line segment using ratios} \\\\\\ A(-3,8)\qquad B(6,-4)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:5} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{5}\implies \cfrac{A}{B} = \cfrac{1}{5}\implies 5A=1B\implies 5(-3,8)=1(6,-4)`

`(\stackrel{x}{-15}~~,~~ \stackrel{y}{40})=(\stackrel{x}{6}~~,~~ \stackrel{y}{-4})\implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-15 +6}}{1+5}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{40 -4}}{1+5} \right)} \\\\\\ C=\left(\cfrac{-9}{6}~~,~~ \cfrac{36}{6} \right)\implies C=\left( -\cfrac{3}{2}~~,~~6 \right)`