Question

If the centroid of the triangle is origin and two of its vertices are (3, –5, 7) and (–1, 7, –6), then the third vertex is ______________________.

Answer 1

Two vertices of triangle are given i.e A(3, –5, 7) and B(–1, +7, –6)
Let w suppose third vertex C(x, y, z)
Since centroid is given by origin i.e O(0, 0, 0)
$$\begin{gathered} \mathrm{Using}\mathrm{formula}\mathrm{coordinates}\mathrm{of}\mathrm{centroid}=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\right) \\ \mathrm{where}\mathrm{vertices}\mathrm{are}\mathrm{triangle}\mathrm{are}\left(x_1,y_1,z_1\right)\left(x_2,y_2,z_2\right)\mathrm{and}\left(x_3,y_3,z_3\right) \end{gathered}$$
Hence $0=\frac{3+\left(-1\right)+x}{3},0=\frac{-5+7+y}{3},0=\frac{7-6+z}{3}$
i.e x = –2, y = –2, z = –1
∴ Third vertex of triangle is given by (–2, –2, –1)