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Set of noncongruent triangles

Source: 2014 AMC 12B #12

February 20, 2014
AMC

Problem Statement

A set S consists of triangles whose sides have integer lengths less than 55, and no two elements of S are congruent or similar. What is the largest number of elements that SS can have?
<spanclass=′latex−bold′>(A)</span>  8<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 12{<span class='latex-bold'>(A)</span>\ \ 8\qquad<span class='latex-bold'>(B)</span>\ 9\qquad<span class='latex-bold'>(C)</span>\ 10\qquad<span class='latex-bold'>(D)</span>}\ 11\qquad<span class='latex-bold'>(E)</span>\ 12
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