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Question
`square`ABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove `square`APCQ is a parallelogram.
Solution
Given: `square`ABCD is a parallelogram. P and Q are the midpoints of sides AB and DC respectively.
To prove: `square`APCQ is a parallelogram.
Proof:
AP = `1/2` AB …(i) [P is the midpoint of side AB]
QC = `1/2` DC …(ii) [Q is the midpoint of side CD]
`square`ABCD is a parallelogram. ...[Given]
∴ AB = DC ...[Opposite sides of a parallelogram]
∴ `1/2` AB = `1/2` DC ...[Multiplying both sides by `1/2`]
∴ AP = QC …(iii) [From (i) and (ii)]
Also, AB || DC ...[Opposite angles of a parallelogram]
i.e. AP || QC …(iv) [A–P–B, D–Q–C]
From (iii) and (iv),
AP = QC
AP || QC
A quadrilateral is a parallelogram if its opposite sides is parallel and congruent.
∴ `square`APCQ is a parallelogram.
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