Question

What are the coordinates of the point on the directed line segment from left bracket, minus, 6, comma, minus, 4, right bracket(−6,−4) to left bracket, 3, comma, minus, 1, right bracket(3,−1) that partitions the segment into a ratio of 1 to 2?

Answer 1

let's say the segment has P(-6 , -4) and R(3 , -1) and is being partitioned by a point Q into a ratio of 1 : 2, where's Q?

`\textit{internal division of a line segment using ratios} \\\\\\ P(-6,-4)\qquad R(3,-1)\qquad \qquad \stackrel{\textit{ratio from P to R}}{1:2} \\\\\\ \cfrac{P\underline{Q}}{\underline{Q} R} = \cfrac{1}{2}\implies \cfrac{P}{R} = \cfrac{1}{2}\implies 2P=1R\implies 2(-6,-4)=1(3,-1)`

`(\stackrel{x}{-12}~~,~~ \stackrel{y}{-8})=(\stackrel{x}{3}~~,~~ \stackrel{y}{-1}) \implies Q=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-12 +3}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-8 -1}}{1+2} \right)} \\\\\\ Q=\left( \cfrac{ -9 }{ 3 }~~,~~\cfrac{ -9}{ 3 } \right)\implies Q=(-3~~,~-3)`