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# Pythagoras for a Tetrahedron

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### Reach for Polydron

### Tetra Inequalities

### Tetra Square

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Age 16 to 18

Challenge Level

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There are a few different approaches you can try.

- You could use Heron's formula for the area of a triangle.

- You could use $\tfrac 1 2 ab \sin C$ for the area of a triangle. It might be helpful to remember that $\cos^2 \theta+\sin^2 \theta = 1$.

- You could try and find a point $X$ on $AB$ such that $CX$ is perpendicular to $AB$, and then use $\tfrac 1 2 bh$ for the area of a triangle. Vectors might be useful here, as well as the fact that ${\bf p}\cdot {\bf q}=0$ if and only if the vectors ${\bf p}$ and ${\bf q}$ are perpendicular. There are quite a lot of perpendicular vectors in this question!

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

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?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.