Question

Find the third vertex of ∆ABC if two of its vertices are B(−3, 1) and C(0, −2) and its centroid is at the origin.

Answer 1

Two vertices of ∆ABC are B(−3,1) and C(0, −2). Let the third vertex be A(a, b).
Then, the coordinates of its centroid are
$$\begin{gathered} \left(\frac{-3+0+a}{3},\frac{1-2+b}{3}\right) \\ \mathrm{i}.\mathrm{e}.\left(\frac{-3+a}{3},\frac{-1+b}{3}\right) \end{gathered}$$
But it is given that the centroid is at the origin, that is G(0, 0). Therefore,
$$\begin{gathered} 0=\frac{-3+a}{3},0=\frac{-1+b}{3} \\ \Rightarrow 0=-3+a,0=-1+b \\ \Rightarrow 3=a,1=b \\ \Rightarrow a=3,b=1 \end{gathered}$$
Therefore, the third vertex of ∆ABC is A(3, 1).