Pythagoras for a Tetrahedron

Age 16 to 18

Challenge Level Yellow star

Consider a right-angled tetrahedron with vertices at

$O(0,0,0)$ ,

$A(a, 0, 0)$ ,

$B(0, b, 0)$ and

$C(0, 0, c)$ .

Let the area of face

$AOB$ be

$P$ , the area of

$BOC$ be

$Q$ and the area of

$COA$ be

$R$ . Also let the slanted face

$ABC$ have area

$S$ .
(

$S$ is not shown on the diagram above!).

Can you prove that

$P^2+ Q^2+ R^2= S^2$ ?

Equivalently: (area $OBC$ ) $^2 +$ (area $OCA$ ) $^2 +$ (area $OAB$ ) $^2 =$ (area $ABC$ ) $^2$ .

Extension

If you enjoyed this question, you might like explore STEP Support Programme Foundation Assignment 5 which asks a question about the perpendicular distance of face $ABC$ from the origin.

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