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P13 [Geometry] - Turkish NMO 1st Round - 2005

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October 26, 2013
geometrytrapezoid

Problem Statement

Let ABCDABCD be an isosceles trapezoid such that its diagonal is 3\sqrt 3 and its base angle is 60∘60^\circ, where AD∥BCAD \parallel BC. Let PP be a point on the plane of the trapezoid such that ∣PA∣=1|PA|=1 and ∣PD∣=3|PD|=3. Which of the following can be the length of [PC][PC]?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 22<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 33<spanclass=′latex−bold′>(E)</span> 7 <span class='latex-bold'>(A)</span>\ \sqrt 6 \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 2 \sqrt 3 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 7
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