Mathematics
In a parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that:
(i) AX = YC.
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram.
Quadrilaterals
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Answer
(i) Given:
ABCD is a parallelogram.
To prove:
AX = YC.
Proof:
We know that, opposite sides of parallelogram are equal.
AB = CD
⇒ AB = CD
⇒ AX = CY (As, X and Y are mid - points of AB and CD respectively)
Hence, AX = YC.
(ii) To prove:
AX is parallel to YC.
Proof:
Opposite sides of parallelogram are equal.
AB is parallel to DC.
⇒ AX is parallel to YC.
Hence, AX is parallel to YC.
(iii) To prove:
AXCY is a parallelogram.
Proof:
AX = YC
And, AX is parallel to YC.
Since, one pair of opposite sides of quadrilateral AXCY are equal and parallel.
Hence, AXCY is a parallelogram.
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