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Question

State true or false:

ABCD is a quadrilateral in which AD=BC. E,F,G and H are the mid-points of AB,BD,CD and AC respectively.
So can we say that EFGH is a rhombus?

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A
True
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B
False
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Solution

The correct option is A True
In ABC,
E is the mid point of AB and H is the mid point of AC
Hence, by mid point theorem,
EHBC and EH=12BC...(I)
Similarly, In BDC,
FGBC and FG=12BC... (II)
In ACD,
GHAD and GH=12AD...(III)
In ABD,
EFAD and EF=12AD...(IV)
Hence, from I and II
EHFG and EH=FG
and from III and IV
GHEF and GH=EF
and EF=FG=GH=EH (Given AD = BC)
EFGH is a parallelogram with all sides equal.
Hence, EFGH is a rhombus

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