Question:
D is the midpoint of side BC of ΔABC and E is the midpoint of BD. If O is the midpoint of AE, Prove that ar(ΔBOE) = (1/8) ar(ΔABC).
Solution:
Given that
D is the midpoint of sides BC of triangle ABC
E is the midpoint of BD and O is the midpoint of AE
Since AD and AE are the medians of triangles, ABC and ABD respectively
∴ ar(ΔABD) = (1/2) ar(ΔABC) ⋅⋅⋅⋅ (1)
∴ ar(ΔABE) = (1/2) ar(ΔABD) ⋅⋅⋅ (2)
OB is the median of triangle ABE
Therefore,
∴ ar(ΔBOE) = (1/2) ar(ΔABE)
From 1, 2 and 3, we have
∴ ar(ΔBOE) = (1/8) ar(ΔABC)
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