Question

The triangle formed by the points $(0,7,10),(–1,6,6),(–4,9,6)$ is

• A. Equilateral triangle
• B. Isosceles triangle
• C. Right angle triangle
• D. Right-angled isosceles

Answer 1

The correct option is D

Right-angled isosceles

Explanation for correct option

Let $A$ be $\left(0,7,10\right)$, $B\left(-1,6,6\right)$ and $C\left(-4,9,6\right)$

Therefore,

$$\begin{gathered} AB=\sqrt{{\left(0+1\right)}^2+{\left(7-6\right)}^2+{\left(10-6\right)}^2} \\ =\sqrt{1+1+16} \\ =3\sqrt{2} \\ BC=\sqrt{{\left(-1+4\right)}^2+{\left(6-9\right)}^2+{\left(6-6\right)}^2} \\ =\sqrt{9+9+0} \\ =3\sqrt{2} \\ AC=\sqrt{{\left(0+4\right)}^2+{\left(7-9\right)}^2+{\left(10-6\right)}^2} \\ =\sqrt{16+4+16} \\ =6 \\ A{C}^2=A{B}^2+B{C}^2 \end{gathered}$$

Therefore the triangle is right angled at $\angle B$ and $AB=BC$

Therefore the triangle is right angled isosceles

Hence $o\mathrm{ption}\left(\mathrm{D}\right)$ is the answer.