geometry

Triangle ABC, AB=8, BC=10, and AC=12, M is the midpoint of AB, and N is the midpoint of BC. What is the length of MN?

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1. Triangle ABC is similar to triangle MBN. Therefore, the corresponding sides are proportional:

MN/AC = BM/AB,
MN/12 = 4/8,
Cross multiply:
8MN = 48,
MN = 6.

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2. By mid point theorem, mn would equal half ac and would be parallel to ac. Thus mn equals 6. The lengths of ab and bc are not required to answer this question. It can be solved by similarity or basic proportionality theorem also.

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