+0

# help

0
140
1

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

#1
+391
+1

$$\Delta DEB$$  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