Question

Show that the points A(0, 1, 2), B(2, -1, 3) and C(1, -3, 1) are vertices of an isosceles right angled triangle.

Answer 1

We have,

$AB=\sqrt{(2-0{)}^2+(-1-1{)}^2+(3-2{)}^2}=\sqrt{4+4+1}=3,$

$BC=\sqrt{(1-2{)}^2+(-3+1{)}^2+(1-3{)}^2}=\sqrt{1+4+4}=3$

and

$CA=\sqrt{(1-0{)}^2+(-3-1{)}^2+(1-2{)}^2}=\sqrt{1+16+1}=3\sqrt{2}$.

Clearly, AB=BC and $A{B}^2+B{C}^2=A{C}^2$.

Hence, triangle ABC is an isosceles right-angled triangle.