Question

Prove that the tetrahedron with vertices at the point O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

Answer 1

The faces of a regular tetrahedron are equilateral triangles

Let us consider $△OAB$

$OA=\sqrt{(0-0{)}^2+(0-1{)}^2+(0-1{)}^2}$

=$\sqrt{2}=\sqrt{2}$ units

$OB=\sqrt{(1-0{)}^2+(0-0{)}^2+(1-0{)}^2}$

=$\sqrt{2}$ units

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

=$\sqrt{2}$ units

Hence, this face is an equilateral triangle.

Similarly, $△OBC,△OAC,△ABC$ all are equilateral triangle.

Hence, the tetrahedron is regular one.