# 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 Inequalitieslive

##### Age 16 to 18Challenge 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 Slicelive

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

### Tetra Perplive

##### Age 16 to 18Challenge 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 Tetrahedronlive

##### Age 16 to 18Challenge 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.