Question

$ABCD$ is the parallelogram and $E$, $F$, $G$ and $H$ are the midpoints of sides $AB$, $BC$, $CD$ and $DA$ respectively. $EF=6$ cm and perimeter of $EFGH$ is $18$ cm. Find the length of diagonal $BD$.

• A. $3$ cm
• B. $6$ cm
• C. $9$ cm
• D. $12$ cm

Answer 1

The correct option is B $6$ cm

We know the midpoints of sides of parallelogram when joined to form quadrilateral, they form a parallelogram
So, $EFGH$ is parallelogram.
So$EF=HGandHE=GF\text{(pair of opposite sides of parallelogram)}$
$EF=HG=6cm$
let $HE=GF=x$
Now,
$\text{Perimeter of EFGH}=EF+FG+GH+HE$
$18=6+6+x+x$
$2x+12=18$
$2x=18-12$
$2x=6$
$x=\frac{6}{2}$
$x=3$ cm

means $HE=3$ cm.
Now drawn diagonal $BD$

In $△ABD$
$H$ and $E$ are midpoints of sides $AD$ and $AB$ respectively so line joining $H$ and $E$ will be half of its parallel side i.e. $BD$ by mid point theorem.

So, $HE=\frac{BD}{2}$
$3=\frac{BD}{2}$
$BD=3\times 2$
$BD=6$ cm

Hence, diagonal $BD=6$ cm.
So, option $\left(b\right)$ correct.