Question

If G(−2, 1) is the centroid of a ∆ABC and two of its vertices are A(1, −6) and B(−5, 2), find the third vertex of the triangle.

Answer 1

Two vertices of ∆ABC are A(1, −6) and B(−5, 2). Let the third vertex be C(a, b).
Then the coordinates of its centroid are
$$\begin{gathered} C\left(\frac{1-5+a}{3},\frac{-6+2+b}{3}\right) \\ C\left(\frac{-4+a}{3},\frac{-4+b}{3}\right) \end{gathered}$$
But it is given that G(−2, 1) is the centroid. Therefore,
$$\begin{gathered} -2=\frac{-4+a}{3},1=\frac{-4+b}{3} \\ \Rightarrow -6=-4+a,3=-4+b \\ \Rightarrow -6+4=a,3+4=b \\ \Rightarrow a=-2,b=7 \end{gathered}$$
Therefore, the third vertex of ∆ABC is C(−2, 7).