Question

Given the midpoint and one endpoint of a line segment, find the other endpoint.

Answer 1

$\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{\frac{5}{8}}~,~\stackrel{y_1}{\frac{27}{8}})\qquad (\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{x+\frac{5}{8}}{2}~~,~~\cfrac{y+\frac{27}{8}}{2} \right)=\stackrel{midpoint}{\left( -\frac{7}{6}~~,~~-\frac{10}{3} \right)}$

$\bf -------------------------------\\\\ \cfrac{x+\frac{5}{8}}{2}=-\cfrac{7}{6}\implies x+\cfrac{5}{8}=-\cfrac{14}{6}\implies x=-\cfrac{14}{6}-\cfrac{5}{8} \\\\\\ x=\cfrac{-14(4)-5(3)}{24}\implies x=\cfrac{-56-15}{24}\implies \boxed{x=-\cfrac{71}{24}}\\\\ -------------------------------\\\\ \cfrac{y+\frac{27}{8}}{2}=-\cfrac{10}{3}\implies y+\cfrac{27}{8}=-\cfrac{20}{3}\implies y=-\cfrac{20}{3}-\cfrac{27}{8} \\\\\\ y=\cfrac{-20(8)-27(3)}{24}\implies y=\cfrac{-160-81}{24}\implies \boxed{y=-\cfrac{241}{24}}$