Question

B is the midpoint of AC and E is the midpoint of BD. If A (-9,-4), C (-1,6), and E (-4,-3), find the coordinates of D

Answer 1

well, we know what A and C are, so let's find that Midpoint firstly.

`~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{-9}~,~\stackrel{y_1}{-4})\qquad C(\stackrel{x_2}{-1}~,~\stackrel{y_2}{6}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ -1 -9}{2}~~~ ,~~~ \cfrac{ 6 -4}{2} \right) \implies \left(\cfrac{ -10 }{2}~~~ ,~~~ \cfrac{ 2 }{2} \right)\implies \stackrel{\textit{\LARGE B}}{(-5~~,~~1)}`

well, we now know what B is, let's find the point D, keeping in mind that their Midpoint is E(-4 , -3)

`~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ B(\stackrel{x_1}{-5}~,~\stackrel{y_1}{1})\qquad D(\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 -5}{2}~~~ ,~~~ \cfrac{ y +1}{2} \right) ~~ = ~~ \stackrel{\textit{\LARGE E}}{(-4~~,~~-3)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{x-5}{2}=-4\implies x-5=-8\implies \boxed{x=-3} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{y+1}{2}=-3\implies y+1=-6\implies \boxed{y=-7}`