Question

If $G$ be the centroid of $\Delta$ABC, where $A(-1,6,6),B(-4,9,6),G(-5/3,22/3,22/3)$ then 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

$A=(-1,6,6)$, $B=(-4,9,6)$, $G=(\frac{-5}{3},\frac{22}{3},\frac{22}{3})$

$C=(x,y,z)$

$G=\frac{x+y+z}{3}$

$\frac{-5}{3}=\frac{-1-4+x}{3}$

$\Rightarrow x=0$

$\frac{22}{3}=\frac{6+9+y}{3}$

$\Rightarrow y=22-15=7$

$\Rightarrow \frac{22}{3}=\frac{6+6+z}{3}$

$z=10$

$\therefore C=(0,7,10)$

$AB=\sqrt{3^2+3^2+0^2}$

$=\sqrt{1}8=3\sqrt{2}$

$BC=\sqrt{4^2+2^2+4^2}=\sqrt{16+4+16}=6$

$AC=\sqrt{1^2+1^2+4^2}=3\sqrt{2}$

$\therefore A{B}^2+A{C}^2=B{C}^2$

$\therefore △ABC$ is a right angled isosceles triangle.