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P01 [Geometry] - Turkish NMO 1st Round - 2001

Source:

April 18, 2014
geometrytrigonometry

Problem Statement

Let A,B,CA,B,C be points on [OX[OX and D,E,FD,E,F be points on [OY[OY such that ∣OA∣=∣AB∣=∣BC∣|OA|=|AB|=|BC| and ∣OD∣=∣DE∣=∣EF∣|OD|=|DE|=|EF|. If ∣OA∣>∣OD∣|OA|>|OD|, which one below is true?
<spanclass=′latex−bold′>(A)</span><span class='latex-bold'>(A)</span> For every XOY^\widehat{XOY},  Area(AEC)>Area(DBF)\text{ Area}(AEC)>\text{Area}(DBF)
<spanclass=′latex−bold′>(B)</span><span class='latex-bold'>(B)</span> For every XOY^\widehat{XOY},  Area(AEC)=Area(DBF)\text{ Area}(AEC)=\text{Area}(DBF)
<spanclass=′latex−bold′>(C)</span><span class='latex-bold'>(C)</span> For every XOY^\widehat{XOY},  Area(AEC)<Area(DBF)\text{ Area}(AEC)<\text{Area}(DBF)
<spanclass=′latex−bold′>(D)</span><span class='latex-bold'>(D)</span> If m(XOY^)<45∘m(\widehat{XOY})<45^\circ then Area(AEC)<Area(DBF)\text{Area}(AEC)<\text{Area}(DBF), and if 45∘<m(XOY^)<90∘45^\circ < m(\widehat{XOY})<90^\circ then Area(AEC)>Area(DBF)\text{Area}(AEC)>\text{Area}(DBF)
<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(E)</span> None of above
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