Question

The points $(1,2,3)$, $(-1,-1,-1)$ and $(3,5,7)$ are the vertices of

• A. an equilateral triangle
• B. an isosceles triangle
• C. a right triangle
• D. None of these

Answer 1

Answer: (D)

Here, the points are $A(1,2,3)$, $B(-1,-1,-1)$ and $C(3,5,7)$.
$\left|AB\right|=\sqrt{2^2+3^2+4^2}=\sqrt{29}$,
$\left|BC\right|=\sqrt{4^2+6^2+8^2}=\sqrt{116}=2\sqrt{29}$
and $\left|CA\right|=\sqrt{2^2+3^2+4^2}=\sqrt{29}$.
So, $|BC|=|AB|+|CA|$
$\Rightarrow$ The points $A$, $B$, $C$ are collinear.