A Shifting Shape

Geometry Level 5

Three of the four sides of a certain quadrilateral have lengths of 3, 4, and 5, and two of its four angles measure 90°. Let A be the sum of all possible distinct areas this quadrilateral could have. What is \(\lfloor A \rfloor\)?

Details and assumptions

  • If more than one such quadrilateral has the same area, only count that area once.
  • The fourth side can have any positive length, and the other two angles can have any measure.
  • If the fourth side has length 0, then that is not a quadrilateral.

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