Question

Find the coordinates of P so that P partitions AB in the ratio 1 to 3 with A(-5, 4) and B(7,-4)

Answer 1

$\textit{internal division of a line segment using ratios} \\\\\\ A(-5,4)\qquad B(7,-4)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:3} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{1}{3}\implies \cfrac{A}{B} = \cfrac{1}{3}\implies 3A=1B\implies 3(-5,4)=1(7,-4)$

$(\stackrel{x}{-15}~~,~~ \stackrel{y}{12})=(\stackrel{x}{7}~~,~~ \stackrel{y}{-4})\implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-15 +7}}{1+3}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{12 -4}}{1+3} \right)} \\\\\\ P=\left( \cfrac{-8}{4}~~,~~\cfrac{8}{4} \right)\implies P=(-2~~,~~2)$