Question

If $ABCD$ is a parallelogram whose diagonals intersect at $O$ and $\Delta$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. $12cm$

Answer 1

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

$ABCD$ is a parallelogram.

∴$BC=AD=6cm$

and $AB=DC=6cm$,

Triangle $BCD$ is equilateral triangle

then $BD=6cm$

Hence $ABCD$ becomes a rhombus.

∴ Diagonals $AC$ and $BD$ bisect each other at right angle.

$\therefore OD=\frac{1}{2}BD=\frac{6}{2}=3cm$

$\Delta OCD$

$(OC{)}^2=C{D}^2-(OD{)}^2=(6{)}^2-(3{)}^2=36-9=27$

$\therefore OC=3\sqrt{3}$

$\therefore AC=2\times OC=2\times 3\sqrt{3}=6\sqrt{3}cm$