Question

Point A is located at (4,9) and point B is located at (12,5) what are the coordinates of the point that partitions the directed line segment AB in a 1:3 ratio? Enter your answer by filling in the boxes. (_,_)

Answer 1

`\textit{internal division of a line segment using ratios} \\\\\\ A(4,9)\qquad B(12,5)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:3} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{3}\implies \cfrac{A}{B} = \cfrac{1}{3}\implies 3A=1B\implies 3(4,9)=1(12,5)`

`(\stackrel{x}{12}~~,~~ \stackrel{y}{27})=(\stackrel{x}{12}~~,~~ \stackrel{y}{5}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{12 +12}}{1+3}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{27 +5}}{1+3} \right)} \\\\\\ C=\left( \cfrac{ 24 }{ 4 }~~,~~\cfrac{ 32}{ 4 } \right)\implies {\Large \begin{array}{llll} C=(6~~,~~8) \end{array}}`