AMC 8 2025 (Problem 24)In trapezoid ABCDABCDABCD, angles BBB and CCC measure 60∘60^\circ60∘ and AB=DCAB=DCAB=DC. The side lengths are all positive integers, and the perimeter of ABCDABCDABCD is 303030 units. How many non-congruent trapezoids satisfy all of these conditions?(A) 0\text{(A)}\;0(A)0(B) 1\text{(B)}\;1(B)1(C) 2\text{(C)}\;2(C)2(D) 3\text{(D)}\;3(D)3(E) 4\text{(E)}\;4(E)4Related TopicsToolkit 67 — When We Have a Trapezoid in a Problem, Draw the AltitudesToolkit 80 — Trigonometric Ratios in a Right TriangleHints (4)Hint 1Perimeter=2y+3x\text{Perimeter}=2y+3xPerimeter=2y+3xHint 22y+3x=302y+3x=302y+3x=30The side lengths are integers, so xxx, yyy, and y+xy+xy+x are integers.Hint 32y+3x=302y+3x=302y+3x=303x=30−2y3x=30-2y3x=30−2ySince 30−2y30-2y30−2y is even, xxx should be even.x=2ax=2ax=2a2y=30−3x=3(10−x)2y=30-3x=3(10-x)2y=30−3x=3(10−x)Since 2y2y2y is a multiple of 333, yyy should be a multiple of 333.y=3by=3by=3bHint 42(3b)+3(2a)=302(3b)+3(2a)=302(3b)+3(2a)=306b+6a=306b+6a=306b+6a=30a+b=5a+b=5a+b=5ba41322314\begin{array}{c|c}b&a\\ \hline4&1\\3&2\\2&3\\1&4\end{array}b4321a1234Final Answer(E) 444
