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# triangle question

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In right triangle ABC, angle B=90 degrees, and D and E lie on AC such that BD is a median and BE  is an altitude. If BD=3 x DE, compute AB/EC.

Dec 18, 2020

#1
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The trick is that you have to find AB and EC in terms of DE. Let DE be x. Then BD is 3x and BE = 2x√2.

As the midpoint of the hypotenuse is center of the circumcircle of a right triangle, AC is the diameter, so BD, AD,DC are all equal and radii of the same circumcircle. Thus, we have AC = 6x. Thus, AE = 4x and EC = 2x. So, AB = sqrt(16x^2 + 9x^2) = 5x, so, AB/EC = 5/2

Dec 18, 2020
#3
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Whoops, I acutally meant to say sqrt(16x^2 + 8x^2) = 2√6, so it is just √6

Pangolin14  Dec 18, 2020
#2
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In the right triangle, ABC, angle B=90 degrees, and D and E lie on AC such that BD is a median and BE  is an altitude. If BD=3 x DE, compute AB/EC.

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The median of a triangle is a line drawn from one of the vertices to the mid-point of the opposite side. In the case of a right triangle, the median to the hypotenuse has the property that its length is equal to half the length of the hypotenuse.

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AD = CD = BD = 3            DE = 1/3(BD) = 1           EC = 2

BE = sqrt(32 - 12) = √8

BC = sqrt(8 + 22) = √12

AB = sqrt(AC2 - BC2) = √24

AB / EC = √24 / 2

Dec 18, 2020
edited by jugoslav  Dec 18, 2020