In right triangle ABC, angle B=90 degrees, and D and E lie on AC such that \(​​​​\overline{BD}\) is a median and \(\overline{BE}\)  is an altitude. If BD=2 x DE, compute \(\frac{AB}{EC}\).

 Aug 21, 2018

\(\Delta DEB\)  is a \(30 - 60 - 90\) triangle, so \(BE = \sqrt{3}(DE)\)\(EC = DE\), and since we are only looking for ratios, we can assign \(DE\) and \(EC\) to \(x\)\(\Delta ABC\) is a \(30 - 60 - 90\) triangle also. \(AC\) is \(4x\) because \(BD\) is a median, therefore splitting \(AC\) into two. Since \(AC = 4x\)\(BC = 2x\). (If you didn't know already, a \(30 - 60 - 90\) triangle's sides are in a ratio of \(1 : \sqrt{3} : 2\).) If \(BC = 2x\), then \(AB = 2\sqrt{3}x\), and \(EC = x\), so your ratio would be \(\frac{2\sqrt{3}x}{x}\), which simplifies to \(2\sqrt{3}\) when you cancel the \(x\)'s.


- Daisy

 Aug 22, 2018

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