# 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):