Question

What are the coordinates of the point on the directed line segment from (1, -5)(1,−5) to (10, 1)(10,1) that partitions the segment into a ratio of 1 to 5?

Answer 1

Check the picture below.

$\textit{internal division of a line segment using ratios} \\\\\\ A(1,-5)\qquad B(10,1)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:5} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{5}\implies \cfrac{A}{B} = \cfrac{1}{5}\implies 5A=1B\implies 5(1,-5)=1(10,1)$

$(\stackrel{x}{5}~~,~~ \stackrel{y}{-25})=(\stackrel{x}{10}~~,~~ \stackrel{y}{1})\implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{5+10}}{1+5}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-25+1}}{1+5} \right)} \\\\\\ C=\left( \cfrac{15}{6}~~,~~\cfrac{-24}{6} \right)\implies C=\left(\cfrac{5}{2}~~,~~-4 \right)$