# Elven Old Traditional Roof Problem?

Geometry Level 5

To solve this problem, I would have to define what "Elven Old Traditional Roof" ("EOTR" for short) is. Firstly, I will say that it is not the same as "Elven Traditional Roof" ("ETR"). That information is rather invaluable.

EOTR is 3D figure, whose base is a rectangle (usually called $$ABCD$$ where $$|AB| > |BC|$$). It is very similar to the pyramid, but unlike such, it doesn't have a peak point, but it has a peak line $$PQ.$$ Both points $$P$$, $$Q$$ are on distance $$H$$ from the base's plane and on the same side of the plane, and the following must be fulfilled to be labelled as EOTR:

• $$|AP| = |DP| = |BQ |= |CQ|$$ (the length of these segments will be labelled as $$S$$) where $$P \neq Q$$ .
• $$|PQ| = |AB| - |BC|$$

The most important property for every EOTR is volume (obviously). So if we know the following:

• $$|PQ| = 5\sqrt{3}$$
• $$\frac{AB}{BC} = 3.5$$
• $$S = 5\sqrt{6}$$

Calculate the most important property of the given "Elven Old Traditional Roof".

Here is one picture that may be helpful in realizing what EOTR is (enlarge here):

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