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?

• A. True
• B. False

Answer 1

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,
$EH\parallel BC$ and $EH=\frac{1}{2}BC$...(I)
Similarly, In $△BDC$,
$FG\parallel BC$ and $FG=\frac{1}{2}BC$... (II)
In $△ACD$,
$GH\parallel AD$ and $GH=\frac{1}{2}AD$...(III)
In $△ABD$,
$EF\parallel AD$ and $EF=\frac{1}{2}AD$...(IV)
Hence, from I and II
$EH\parallel FG$ and $EH=FG$
and from III and IV
$GH\parallel EF$ and $GH=EF$
and $EF=FG=GH=EH$ (Given AD = BC)
$\therefore$ EFGH is a parallelogram with all sides equal.
Hence, EFGH is a rhombus