Trapezints are those special trapezia whose sides have integer lengths. The parallel sides are of unequal length. Let a be the length of the shorter of the parallel sides and b,c,d the lengths of the other sides in clockwise order. We write [a,b,c,d] as the name of the trapezint. For example [1,3,5,4] is a trapezint with perimeter 13.

a. What is the shorter perimeter that a trapezint can have? Explain why there is not a smaller one.

b. What is the smallest perimeter of a trapezint with a non-parallel sides having different lengths? Explain why there is not a smaller one.

c. What is the smallest perimeter of a trapezint with at least one angle a right angle? Explain why there is not a smaller one.

d. Find all trapezints with perimeter 9.

Note: here we regard two trapezints as different only if they are not congrunt. In particular [a,b,c,d] and [a,d,c,b] are the same.