Question

If two vertices of an equilateral triangle are $(2,1,5)$ and $(3,2,3)$, then its third vertex is:

• A. $(1,2,4)$
• B. $(4,0,4)$
• C. $(0,-4,4)$
• D. $(4,4,1)$

Answer 1

The correct option is B $(4,0,4)$

Let the coordinates of third vertex be $(x,y,z)$

The distance of third vertex from above two vertices are same

So, After distance form between $(2,1,5)and(x,y,z)$ {and equate it with distance between $(3,2,3)and(x,y,z)$. After simplification we get

$\Rightarrow x^2+y^2+z^2-4x-2y-10z+30=x^2+y^2+z^2-6x-4y-6z+22$

$\Rightarrow x+y-2z+4=0$

Among given options , only $(4,0,4)$ satisfies above equation

Therefore the correct option is $B$