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Question

In the given figure, ABCD is a trapezium. M and N are mid points of diagonals AC and BD respectively. Which of the following is true?

A
DC = 2 EB
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B
MN || AB || CD
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C
NE = 0.5 CN
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D
DN = 2 BN
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Solution

The correct option is B MN || AB || CD
Let ABCD be a trapezium in which AB || DC and let M and N be the midpoints of the diagonals AC and BD, respectively.

Construction: Join CN and produce it to meet AB at E.

In ΔCDN and ΔEBN, we have

DN = BN [since N is the midpoint of BD]

DCN=BEN [alternate interior angles]

and CDN=EBN [alternate interior angles]

ΔCDNΔEBN [by AAS congruence rule]

DC = EB and CN =NE [by CPCT rule]

Thus, in ΔCAE, the points M and N are the midpoints of AC and CE, respectively.

MN || AE [by mid-point theroem]

MN || AB || CD


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