Question

If the centroid of the triangle whose vertices are (2, 4), (3, a), (4, 2) is (a, 3), then a = _________.

Answer 1

$$\begin{gathered} \mathrm{Centroid}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{with}\mathrm{vertices}\left(x_1,y_1\right),\left(x_2,y_2\right)\mathrm{and}\left(x_3,y_3\right)\mathrm{is}\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right). \\ \mathrm{Centroid}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{with}\mathrm{vertices}\left(2,4\right),\left(3,a\right)\mathrm{and}\left(4,2\right)=\left(a,3\right) \\ \Rightarrow \left(\frac{2+3+4}{3},\frac{4+a+2}{3}\right)=\left(a,3\right) \\ \Rightarrow \left(\frac{9}{3},\frac{6+a}{3}\right)=\left(a,3\right) \\ \Rightarrow \left(3,\frac{6+a}{3}\right)=\left(a,3\right) \\ \Rightarrow 3=a \end{gathered}$$

Hence, a = 3.