Question

The co-ordinates of the vertices of a triangle are (4, -3),(-5, 2) and ( x, y). If the centroid of the triangle is at the origin, Show that x= y = 1?

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{centroid}G\left(x,y\right)\mathrm{of}\mathrm{a}∆\mathrm{having}\mathrm{its}\mathrm{vertices}\mathrm{at}\left(x_1,y_1\right);\left(x_2,y_2\right)\mathrm{and}\left(x_3,y_3\right)\mathrm{is}\mathrm{given}\mathrm{by} \\ x=\frac{x_1+x_2+x_3}{3},y=\frac{y_1+y_2+y_3}{3} \\ \mathrm{Let}\mathrm{A}\left(4,-3\right);\mathrm{B}\left(-5,2\right);\mathrm{C}\left(x,y\right)\mathrm{be}\mathrm{the}\mathrm{vertices}\mathrm{of}∆\mathrm{ABC}. \\ \mathrm{Now},\mathrm{O}\left(0,0\right)\mathrm{is}\mathrm{the}\mathrm{centroid}\mathrm{of}∆\mathrm{ABC}. \\ \mathrm{Now}, \\ 0=\frac{4+\left(-5\right)+x}{3}\mathrm{and}0=\frac{-3+2+y}{3} \\ \Rightarrow x-1=0\mathrm{and}y-1=0 \\ \Rightarrow x=1\mathrm{and}y=1 \\ \mathrm{So},x=y=1 \end{gathered}$$