X is the mid-point of the side BC of a parallelogram ABCD. DX meets AC in Y Prove that $a r (Δ A D Y) = 2 a r (D Y C)$

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Solution

The $Δ$ AYD and $Δ$ XYX can be proved similar by AAA similarity criteria.
Also, AD = 2 XC as X is the midpoint of BC.
Hence, AY: TC = 2: 1 (As ratio is same for the similar triangles)
Hence, $a r (Δ A D Y) = 2 a r (D Y C)$ = 2: 1 [Triangles on the same base and have same height have area in the ratio of bases.]

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