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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.
problem
Tetra inequalities
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?
problem
Tetra slice
Age
16 to 18
Challenge level
Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?
problem
Favourite
Tetra perp
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.
problem
Favourite
Pythagoras for a tetrahedron
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.