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Turkey NMO 2010 1st Round - P25 (Geometry)

Source:

July 26, 2012
geometry

Problem Statement

Let PP and QQ be points on the plane ABCABC such that m(BAC^)=90∘m(\widehat{BAC})=90^\circ, ∣AB∣=1|AB|=1, ∣AC∣=2|AC|=\sqrt 2, ∣PB∣=1=∣QB∣|PB|=1=|QB|, ∣PC∣=2=∣QC∣|PC|=2=|QC|, and ∣PA∣>∣QA∣|PA|>|QA|. What is ∣PA∣/∣QA∣|PA|/|QA|?
<spanclass=′latex−bold′>(A)</span> 2+3<spanclass=′latex−bold′>(B)</span> 5−6<spanclass=′latex−bold′>(C)</span> 6−2<spanclass=′latex−bold′>(D)</span> 6+1<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ \sqrt 2 +\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ 5-\sqrt 6 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 6 -\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 6 + 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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