# Do degenerate triangles count? (2014 AMC 12B #12)

The problem is this: A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?

I got an answer of 9, but the solution on aops says: "Based on the wording of Problem 13 to specifically exclude triangles with zero area: "... triangle with positive area ...", the definition of a triangle in this test includes degenerate ones. That is, the triangle inequality is not strict."

They are then able to find three degenerate triangles, changing the answer to 12 which is also an answer choice. Can degenerate triangles be assumed to count as triangles in this way? Shouldn't they specify within the problem?

• I'm confused: Title says #12, description says problem 13. Also I think you answered your own question. – qwr Feb 21 '14 at 5:46
• No, the solution to number 12 on aops was referencing the wording of problem 13 which also dealt with degenerate triangles. Besides, problem 13 excludes triangles with zero area, and the solution is saying that problem 12 is therefore including them by not specifying. – user130413 Feb 21 '14 at 5:47
• Opinion, not answer: If the word triangle is used in a contest problem, the natural interpretation is the commonsense meaning. What wording another question used is irrelevant. – André Nicolas Feb 21 '14 at 5:54
• ...and how did they get three degenerate triangles anyway? They could have sides $4,3,1$ or $4,2,2$ or $3,2,1$ or $2,1,1$ so that's four. Perhaps they argue that the first two are the same; but then surely you would have to say that all those I have just mentioned are similar, so it's only one. And then do you allow a "triangle" with sides $0,0,0$? When you come down to it, unless you make an assumption like that suggested by André, it's just a very poorly posed problem. – David Feb 21 '14 at 6:03
• @David, $(4,2,2)$ and $(2,1,1)$ are similar. But the problem is poorly posed indeed. – sas Feb 21 '14 at 6:17

## 1 Answer

How is similarity for degenerate triangles determined? One may assume that a deg. triangle is a line segment, so they are all similar. Or are they different, in which case the side ratios have to be considered? How will MAA deal with this problem? Degenerate Triangles are triangles, after all, and if commonsense was sufficient for #13, the writers would not have written positive area, as that would be implied. EDIT: I looked up degenerate triangles, and it seems that the similarity depends on the RATIO of the sides: A-----B-----------C could be a degenerate triangle, but it is not the same as A--------B---------C, even though the lengths are the same. So therefore, 422 and 112 are similar, while the other 2 are not: there are 3 total, with 12 triangles total. However, MAA will likely accept both 12 and 9, as the question was poorly worded