Q9 of 25 Page 9

In a Δ ABC, if L and M are points on AB and AC respectively such that LM||BC. Prove that:

(i) ar(Δ LCM) = ar(Δ LBM)


(ii) ar(Δ LBC) = ar(Δ MBC)


(iii) ar(Δ ABM) = ar(Δ ACL)


(i) Clearly, triangles LMB and LMC are on the same base LM and between the same parallels LM and BC.


Therefore,


Area ( = Area () …. (1)


(ii) We observe that triangles LBC and MBC are on the same base BC and between the same parallels LM and BC.


Therefore,


Area (ΔLBC) = Area (ΔMBC) ….. (2)


(iii) We have,


Area ( = Area ( [From (1)]


Area ( + Area ( = Area ( + Area (


Area ( = Area (


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7

ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove that

(i) ar(Δ ADO) = ar(Δ CDO)


(ii) ar(Δ ABP) = ar(Δ CBP)

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ABCD is a parallelogram. E is a point on BA such that BE = 2 EA and F is a point on DC, such that DF = 2FC. Prove that AECF is a parallelogram whose area is one third of the area of parallelogram ABCD.

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ABCD is a trapezium in which AB||DC. DC is produced to E such that CE=AB, Prove that ar(Δ ABD) = ar(Δ BCE)

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If P is any point in the interior of a parallelogram ABCD, then prove that area of the triangle APB is less than half the area of parallelogram.