Quandaries and Queries


This is a 9th grade geometry problem:
In triangle ABC, AB=4, AC=6 and AD=5, where D is the midpoint of BC. Determine BC.


Hi Amy,

There is a formula known as Stewart's theorem that deals with a more general problem. Let's use the standard notation: a = BC, b = AC, c = AB, and let m = AD = our median from A. Then Stewart's theorem tells us that

m2 = [2(b2 + c2) - a2]/4.

This formula comes from applying the cosine law to the triangle ABD, then replacing cos B by (a2 + c2 - b2)/2ac.

[For the record, Stewart's theorem tells us that if the point D divids BC into segments of length r = BD and s = DC, and AD has length p, then

a(p2 + rs) = rb2 + sc2.

Here in your problem, r = s = a/2.]




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