question_answer

If ABCD is a parallelogram whose diagonals intersect at O and \[\Delta \,ABC\] 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

Correct Answer: B

Solution :

  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 \[\Delta \,\,OCD,\,O{{C}^{2}}=C{{D}^{2}}-O{{D}^{2}}\]                 \[=36-9=27\] \[\therefore \]  \[OC=3\sqrt{3}\] and        \[AC=2.\,\,OC=6\sqrt{3}\]