Question

If $A=(-1,6,6)$ , $B=(-4,9,6)$ , $G=\frac{1}{3}(-5,22,22)$ and $G$ is the centroid of the $\Delta ABC$ then the name of the triangle $ABC$ is

• A. an isosceles triangle
• B. a right angled triangle
• C. an equilateral triangle
• D. a right-angled isosceles triangle

Answer 1

The correct option is D a right-angled isosceles triangle

The centroid of a triangle if given by
$G=\left(\right(\frac{x_1+x_2+x_3}{3}),\frac{y_1+y_2+y_3}{3}),\frac{z_1+z_2+z_3}{3}\left)\right)$
Let the coordinates of C be $(x,y,z)$; then
$G=(\frac{-5}{3},\frac{22}{3},\frac{22}{3})=(\frac{x-1-4}{3},\frac{y+6+9}{3},\frac{z+6+6}{3})$
Comparing LHS and RHS gives us
$x-5=-5$ or $x=0$; similarly $y+15=22$ or $y=7$. and $z+12=22$ or $z=10$.
Hence
$C=(0,7,10)$.
Thus the coordinates of the vertices of the triangle are
$A(-1,6,6)$ $B=(-4,9,6)$ and $C(0,7,10)$.
Now
$AB=3\sqrt{2}$
$BC=6$
$AC=3\sqrt{2}$
Now
$A{B}^2+A{C}^2=(3\sqrt{2}{)}^2+(3\sqrt{2}{)}^2=18+18=36$
$=B{C}^2$.
Hence it is a right angled isosceles triangle right angled at A