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*DE, compute \(\frac{AB}{EC}.\).

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ant101 Mar 13, 2019

#1**+1 **

Since BD is a median....then AD = DC........and we can construct a circle with radius DC = 2

And since angle B is right and A and C lie on the diameter endpoints .....then right triangle ABC can be inscribed in the circle

Then BD is a radius = 2.....and...in right triangle BDE, DE = (1/2) BD = 1 ....and BE = √3

And in right triangle BDE, angle DBE = 30° and angle EDB = 60°

Then angle ADB is supplemental to EDB = 120°

And AD = DB.....so in triangle ABD, angles BAD and ABD = 30°

Then, in right triangle ABC, angle ACB = 60°

So....in right triangle ABC....BC = (1/2) AC = (1/2)(4) = 2 and AB = 2√3

And in right triangle BCE....angle EBC = 30°....so....EC = 1/2 BC = (1/2)(2) = 1

So

AB / EC = [2√3] / [1] = = 2√3

CPhill Mar 13, 2019