A Quartet of Tetrahedra

This feature is all about tetrahedra, with four problems for you to solve.

You can find another problem involving a tetrahedron in STEP Support Programme Foundation Assignment 5.

The last day for submitting solutions to the live problems is Monday 12 December.

 

 

Tetra Inequalities live

Age 16 to 18
Challenge Level
Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

Tetra Slice live

Age 16 to 18
Challenge Level
Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?

Tetra Perp live

Age 16 to 18
Challenge Level
Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.

Pythagoras for a Tetrahedron live

Age 16 to 18
Challenge Level
In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.