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Turkey NMO 2010 1st Round - P17 (Geometry)

Source:

July 26, 2012
geometry

Problem Statement

Let A,B,C,DA,B,C,D be points in the space such that ∣AB∣=∣AC∣=3|AB|=|AC|=3, ∣DB∣=∣DC∣=5|DB|=|DC|=5, ∣AD∣=6|AD|=6, and ∣BC∣=2|BC|=2. Let PP be the nearest point of BCBC to the point DD, and QQ be the nearest point of the plane ABCABC to the point DD. What is ∣PQ∣|PQ|?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 372<spanclass=′latex−bold′>(C)</span> 57211<spanclass=′latex−bold′>(D)</span> 922<spanclass=′latex−bold′>(E)</span> 22 <span class='latex-bold'>(A)</span>\ \frac{1}{\sqrt 2} \qquad<span class='latex-bold'>(B)</span>\ \frac{3\sqrt 7}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{57}{2\sqrt{11}} \qquad<span class='latex-bold'>(D)</span>\ \frac{9}{2\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ 2\sqrt 2
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