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Q.

State whether the statement is true or false.
In a parallelogram ABCD, BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, then ar (ΔBPC) = ar (ΔDPQ).

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a

True

b

False

answer is A.

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Detailed Solution

Given that, in a parallelogram ABCD, BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P.
Given that,ABCD is a parallelogram and BC is produced to a point Q.
AD = CQ.
If AQ intersects DC at P, then ar (ΔBPC) = ar (ΔDPQ).
Question Image

Question Image

Since ABCD is a parallelogram so, AB|| CD.

⇒ A B | | P C

AD|| BC

⇒ A D | | C Q

The triangles between the same parallel lines and on the common base, have equal areas. So,

a r (Δ A P C) = a r (Δ P B C)

. [1]
We are given that AD = QC and AD ||QC. So,

a r (Δ D C Q) = a r (Δ A C Q) On subtracting ar(Δ PCQ) from both sides, we get: a r (Δ D C Q) − a r (Δ P C Q) = a r (Δ A C Q) − a r (Δ P C Q) a r (Δ D P Q) = a r (Δ A P C) [2]

From [1] and [2] we get area(ΔBPC) = area(ΔDPQ).
Hence, the given statement is true and the correct option is 1.

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State whether the statement is true or false.In a parallelogram ABCD, BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, then ar (ΔBPC) = ar (ΔDPQ).