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):

Elven Old Traditional Roof, not the one from the problem though

Elven Old Traditional Roof, not the one from the problem though

If You have any question related to Elven 3D objects, such as this one, or about the problem, feel free to ask it here.
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