Question

What are the coordinates of the point on the directed line segment from (7,9) to (7,-3) that partitions the segment into a ratio of 2 to 1?

Answer 1

`\textit{internal division of a line segment using ratios} \\\\\\ A(7,9)\qquad B(7,-3)\qquad \qquad \stackrel{\textit{ratio from A to B}}{2:1} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{2}{1}\implies \cfrac{A}{B} = \cfrac{2}{1}\implies 1A=2B\implies 1(7,9)=2(7,-3)`

`(\stackrel{x}{7}~~,~~ \stackrel{y}{9})=(\stackrel{x}{14}~~,~~ \stackrel{y}{-6}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{7 +14}}{2+1}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{9 -6}}{2+1} \right)} \\\\\\ C=\left( \cfrac{ 21 }{ 3 }~~,~~\cfrac{ 3}{ 3 } \right)\implies {\Large \begin{array}{llll} C=(7~~,~~1) \end{array}}`