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AMC 12 A #0kj6

AMC 12 AHard

From module Angle Chasing Mastery

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
Cartesian coordinatesVectorsAngle chasing

Problem

Let ABCDABCD be an isosceles trapezoid with BC‾∥AD‾\overline{BC} \parallel \overline{AD} and AB=CDAB = CD. Points XX and YY lie on diagonal AC‾\overline{AC} with XX between AA and YY, as shown in the figure. Suppose ∠AXD=∠BYC=90∘\angle AXD = \angle BYC = 90^\circ, AX=3AX = 3, XY=1XY = 1, and YC=2YC = 2. What is the area of ABCDABCD? figure (A) 15 (B) 5115\sqrt{11} (C) 3353\sqrt{35} (D) 18 (E) 777\sqrt{7}

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