Question

What are the coordinates of the point on a directed line segment from (-8,8) to (-2,-10) that partitions the segment into a ratio of 1 to 2

Answer 1

`\textit{internal division of a line segment using ratios} \\\\\\ A(-8,8)\qquad B(-2,-10)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{2}\implies \cfrac{A}{B} = \cfrac{1}{2}\implies 2A=1B\implies 2(-8,8)=1(-2,-10)`

`(\stackrel{x}{-16}~~,~~ \stackrel{y}{16})=(\stackrel{x}{-2}~~,~~ \stackrel{y}{-10}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-16 -2}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{16 -10}}{1+2} \right)} \\\\\\ C=\left( \cfrac{ -18 }{ 3 }~~,~~\cfrac{ 6}{ 3 } \right)\implies {\Large \begin{array}{llll} C=(-6~~,~~2) \end{array}}`