Question

ABC is a triangle whose vertices are A(3,4), B(-2,-1) and C(5,3). If G is the centroid and BDCG is a parallelogram then find the coordinates of the vertex D.

Answer 1

Centroid $G=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)G=\left(\frac{3-2+5}{3},\frac{4-1+3}{3}\right)=\left(2,2\right)$

In parallelogram diagonals bisect each other.
so, Midpoint of BG = Midpoint of CD
i.e, $\left(\frac{x+2}{2},\frac{y+2}{2}\right)=\left(\frac{-2+5}{2},\frac{-1+3}{2}\right)\left(\frac{x+2}{2},\frac{y+2}{2}\right)=\left(\frac{3}{2},1\right)$
$\frac{x+2}{2}=\frac{3}{2}\Rightarrow x+2=3\Rightarrow x=1\frac{y+2}{2}=1\Rightarrow y=0$The coordinate of vertex D = (1,0)