Question

point j has coordinate (-2,8) and point r has coordinates (-7,-3. find the coordinate of point A that partition line segment JR in the ratio of 5:2

Answer 1

So, the point A cuts jr into a 5:2 ratio, where the pieces of jA and Ar are at a 5:2 ratio, thus

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ j(-2,8)\qquad r(-7,-3)\qquad \qquad \stackrel{\textit{ratio from \underline{j} to \underline{r}}}{5:2} \\\\\\ \cfrac{j\underline{A}}{\underline{A} r} = \cfrac{5}{2}\implies \cfrac{j}{r} = \cfrac{5}{2}\implies 2j=5r\implies 2(-2,8)=5(-7,-3)\\\\ -------------------------------$

$\bf A=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\ -------------------------------\\\\ A=\left(\cfrac{(2\cdot -2)+(5\cdot -7)}{5+2}\quad ,\quad \cfrac{(2\cdot 8)+(5\cdot -3)}{5+2}\right) \\\\\\ A=\left(\cfrac{-4-35}{7}~~,~~\cfrac{16-15}{7} \right)\implies A=\left( -\cfrac{39}{7}~~,~~\cfrac{1}{7} \right) \\\\\\ A=\left( -5\frac{4}{7}~~,~~\cfrac{1}{7} \right)$