Originally posted by phgao
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 ...[text shortened]... as different only if they are not congrunt. In particular [a,b,c,d] and [a,d,c,b] are the same.
a. 5 (side lengths 1,1,1,2). You can't get shorter, as [1,1,1,1] is not allowed because parallel sides must have unequal length.
b. 6 (side lengths 1,1,2,2). You can't get shorter as each pair of opposite sides has unequal length, so one of each pair must have length >1.
c. 14, in the form [1,5,5,3], [1,5,4,4] or a mirror image: If we have one right angle between a parallel side P and a non-parallel side Q, we must have another between the same non-parallel side R and the other parallel. That forms a 'U' shape, only the sides of the U can't be of equal height. By Pythagoras, the length of the remaining side S is given by S = Q^2 + (P-R)^2. The smallest solution to this in positive integers is S = 5, with one of:
Q = 3, P = 5, R = 1
Q = 3, P = 1, R = 5
Q = 4, P = 4, R = 1
Q = 4, P = 1, R = 4
d. No side can be longer than 4 by the triangle inequality; the parallel sides cannot be the same length; and mirror images are considered equivalent. Hence an equivalence class of trapezints with perimeter 9 can be indicated by two unordered pairs of integers between 1 and 4 whose sum is 9: the first pair represents the parallel sides, which must be unequal, and the second pair the other sides. These equivalence classes are as follows: