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State whether the following statement is true or false:

$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides AB and CD, see figure. If AQ intersects DP in $S$ and BQ intersects CP in $R$ ,then APCQ is a parallelogram.

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True

False

answer is A.

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Detailed Solution

Given, ABCD is a parallelogram with P, Q as midpoints of AB and CD.
Question Image

In quadrilateral APCQ,

A B ∥ C D

⇒ A P | | Q C

A P = 12 A B, C Q = 12 D C

[

P, Q are midpoints of AB, CD]

Now,

A B = C D

[Opposite sides of parallelogram]

⇒ A P = Q C

[From the figure]
Thus,

A P | | Q C, A P = Q C

.
Therefore, it is true that APCQ is a parallelogram.
Hence, the correct option is 1.

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