Question

Point A has coordinates (-4,-2). Point B has coordinates (6,3). Find the coordinates of point P that partition AB in the ratio 3:2

Answer 1

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

`(\stackrel{x}{-8}~~,~~ \stackrel{y}{-4})=(\stackrel{x}{18}~~,~~ \stackrel{y}{9})\implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-8+18}}{3+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-4+9}}{3+2} \right)} \\\\\\ P=\left( \cfrac{10}{5}~~,~~\cfrac{5}{5} \right)\implies P=(2~~,~~1)`