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P25 [Geometry] - Turkish NMO 1st Round - 2005

Source:

October 26, 2013
geometryrectangle

Problem Statement

Let EE, FF, GG be points on sides [AB][AB], [BC][BC], [CD][CD] of the rectangle ABCDABCD, respectively, such that ∣BF∣=∣FQ∣|BF|=|FQ|, m(FGE^)=90∘m(\widehat{FGE})=90^\circ, ∣BC∣=43/5|BC|=4\sqrt 3 / 5, and ∣EF∣=5|EF|=\sqrt 5. What is ∣BF∣|BF|?
<spanclass=′latex−bold′>(A)</span> 10−22<spanclass=′latex−bold′>(B)</span> 3−1<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 11−32<spanclass=′latex−bold′>(E)</span> 1 <span class='latex-bold'>(A)</span>\ \dfrac{\sqrt{10} - \sqrt{2}}{2} \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 -1 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{\sqrt{11} - \sqrt{3}}{2} \qquad<span class='latex-bold'>(E)</span>\ 1
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