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XY is a line parallel to side BC of a t...

XY is a line parallel to side BC of a triangle ABC. If `B E\ ||\ A C`and `C F\ ||\ A B`meet XY at E and F respectively, show that `a r\ (A B E)\ =\ a r\ (A C F)`

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Draw prependiculars GM and DN from C and D respectively to AB.
Given that
area of `Delta AOD = " area of "DeltaBOC`
Adding area of `Delta AOB` on both sides.
area of `Delta AOD + " area of " Delta AOB`
= area of `DeltaBOC + " area of "DeltaAOB`
`rArr` area of `DeltaABD = " area of "DeltaABC`
`rArr` area of `DeltaABD = " area of "DeltaABC`
`rArr (1)/(2) xx AB xx DN = (1)/(2) xx AB xx CM`
`rArr DN = CM`

`:. DC||AB`
(perpendicular distancce between two parallel lines area always same)
`:. square ABCD` is a trapezium.
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