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# Triangles!

Nick George
Ranch Hand
Posts: 815
A pretty straight forward, but interesting, one regarding a traingle as follows:

In triangle ABC, AB=4, AC=6, AD=5, where D is the midpoint of BC. What is BC?

Eric Pascarello
author
Rancher
Posts: 15385
6
c1^2 + c2^2 = 5^2 + 5^2 - 4^2 - 6^2

Arjun Shastry
Ranch Hand
Posts: 1901
1
Cosine rule for two triangles:ABD and ABC
cos B = a**2+c**2-b**2/2*a*c
gives BC = 2.
[ June 05, 2004: Message edited by: Ram Abdullah D'Souza ]

Nick George
Ranch Hand
Posts: 815
Well, that looks like the right answer, howerver, i am a bit lost as to how it is obtained. Using law of cosines, I was able to get the following three equations from the three triangles:

s=length of BD (half of BC)
beta1+beta2=angle BAC

4s�=52-48cos(beta1+beta2)

s�=41-40cos(beta1)

s�=61-60cos(beta1)

after solving all three together, i found s to be something like 1.223.

can i have a little more explanation as to how you came at your answers?

Arjun Shastry
Ranch Hand
Posts: 1901
1
We can apply cosine rule only to two triangles.namely,ABD and ABC.
Let 2x be length of BC.then BD=DC = x.
For triangle,ABD,
cosB = (4*4+x*x-5*5)/2*4*x-----------(1)
For triangle ,ABC
cosB = (4*4+(2x*2x)-6*6)/2*4*2x--------(2)
Equating 1 &2 ,will give x=1-->BC=2

Nick George
Ranch Hand
Posts: 815
quite understood... Thanks!