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Question
In triangle ABC, D and E are mid-points of the sides AB and AC, respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.
Sum
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Solution
- Since D and E are the midpoints of sides AB and AC, segment DE is parallel to BC by the Midpoint Theorem, which means DE || BF.
- Since EF is drawn parallel to AB (or DB) through point E, we have EF || DB.
- Since both pairs of opposite sides are parallel (DE || BF and EF || DB), quadrilateral BDEF is a parallelogram.
BD = EF = `1/2"AB" = 1/2 xx 8` = 4 cm
BF = DE = `1/2"BC" = 1/2 xx 9` = 4.5 cm
Therefore, perimeter of BDEF = 2(BF + EF) = 2(4.5 + 4) = 17 cm.
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