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11 sides
The diagonal of a polygon is a line segment that joins two non adjacent vertices. So for a n-sided polygon, each vertex can be joined to n-3 other sides. So there ought to be n x (n-3) diagonals for a n-sided polygon; however, each diagonal is counted twice, once by the vertex at each end. So the number of diagonals is:
diagonals = n (n-3) / 2
So for 44 diagonals:
44 = n (n-3) / 2
88 = n2 - 3n
n2 - 3n - 88 = 0
(n - 11)(n + 8) = 0
n = 11 or -8
Since the number of sides can't be negative, the number of sides is 11, a hendecagon (or undecagon)
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What shape has equal diagonals that bisect each other but no other information is known?
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