IBM Ponder This #150 · October 2010

In the following story, can you fill in the values a, b, c, d,and e with integer numbers so as to make the story true? And what would be X?

Ponder: What is the area of the lake?
This: I can not tell since I do not know anything about it.
Ponder: It has the shape of a convex pentagon; its edges lengths (in kilometers) are integers.
This: Can you give me more details?
Ponder: Here are the integer lengths of the edges: a, b, c, d,and e. I sorted them for you: a<b<c<d<e.
This: Now we are getting somewhere, but I still do not know the area.
Ponder: Last hint: Two of the angles are right (90 degrees).
This: Thank you. The lake area MUST be X square kilometers.

Update: October 4th: X, the area of the lake, does not have to be an integer.

Update: October 5th: Please specify, in your solution, the order of the edges.

Solution

Best opened after a real attempt

To be added.