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X and Y are points on the side LN of the...

X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (see figure). Prove that `ar (DeltaLZY) = ar (MZYX)`.

Text Solution

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`trianglePSM and triangleQSM` have the same base SM and lie between the same parallels PQ and MS.
`therefore ar(trianglePSM)=ar(triangleQSM)`
`rArr ar(trianglePSM)+ar(triangleMSN)=ar(triangleQSM)+ar(triangleMSN)`
`rArr ar(trianglePSN)=ar(QSNM)`.
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