<s>[INST] The area of a peculiar triangle is given to be $10$ square units.  Additionally, the triangle has three unusual properties:
(P1) The length of the shortest altitude is an integer.
(P2) Exactly two sides of the triangle are of integral length.
(P3) If the shortest altitude of the triangle is $a$, the ratio of the lengths of this altitude to the altitude to the hypotenuse is $\frac{a}{2}$.
What is the length of the hypotenuse of this triangle?[/INST]