Problem: In the adjoining figure triangle ABC is such that AB=4 and AC=8. If M is the midpoint of BC and AM=3, what is the length of BC?
Answer Choices:
A. 226​
B. 4+213​
C. 231​
D. not enough information
E. 9 given to solve the problem
Solution:
In the adjoining figure let h be the length of altitude AN drawn to BC, let x=BM and let y=NM. Then
h2+(x+y)2h2+y2h2+(x−y)2​=64=9=16​
Subtracting twice the second equation from the sum of the first and third equations yields 2x2=62. Thus x=31​ and BC=231​.
OR
Applying the parallelogram law to the parallelogram having AB and AC as adjacent sides yields
42+82​=2(32)+2x2x​=31​.​
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