Question

If $ABCD$ is a parallelogram whose diagonals intersect at $O$ and $BCD$ is an equilateral triangle having each side of length $6$ cm, then the length of diagonal $AC$ is :

• A. $3\sqrt{3}$ cm
• B. $6\sqrt{3}$ cm
• C. $3\sqrt{6}$ cm
• D. $12$ cm

Answer 1

The correct option is B $6\sqrt{3}$ cm

$ABCD$ is a parallelogram
$\therefore BC=AD=6$ cm
And $AB=DC=6$ cm
Hence, $ABCD$ becomes a rhombus.
$\therefore$ Diagonals $AC$ and $BD$ bisect each other at right angle.

$\therefore OD=\frac{1}{2}BD=3$ cm
From $\Delta OCD$,

$O{C}^2=C{D}^2-O{D}^2$ .... Pythagoras theorem

$=36-9$

$=27$

$\therefore OC=3\sqrt{3}$
and $AC=2\cdot OC=6\sqrt{3}$