+0  
 
0
111
3
avatar

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
avatar+419 
0

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
avatar+419 
0

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

Pangolin14  Dec 18, 2020
 #2
avatar+1162 
+3

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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

53 Online Users

avatar
avatar
avatar
avatar
avatar
avatar