Pyramid problem - find the height of the pyramid by given edges and a side from the base

Question

The base of a pyramid is a right triangle with the longest side = 12cm. All non base edges are also equal to 12cm. And how can I find the height of the pyramid?

Answer 1

Use Thales' theorem.

You can select the points

$$ A = (-6,0,0), B = (+6,0,0), $$

to form the longest side of the base triangle. The other point $C$ lies on a circle of which $AB$ is the diameter, as

$$ y^2 + \big[ 6 - x \big]^2 + y^2 + \big[ 6 + x \big]^2 = 144, $$

so

$$ x^2 + y^2 = 36 $$

The top point $T$ is given by $(0,0,z)$ and it is clear that

$$ |AT| = |BT| = |CT| $$

as $A$, $B$ and $C$ lie on the same circle, as $|AT| = |BT| = |AB|$, we obtain

$$ z = 6 \sqrt{3} $$