Question

Choose the correct answer form alternatives given.
One of the four angles of a rhombus is 60. If the length of each side of the rhombus is 8 cm. then the length of the longer diagonal is

• A. $8\sqrt{3}$ cm
• B. 8 cm
• C. $4\sqrt{3}$ cm
• D. $\frac{8}{\sqrt{3}}$ cm

Answer 1

The correct option is A $8\sqrt{3}$ cm

Let ABCD be the rhombus with $\angle A={60}^{\circ }$,
Hence $\angle ={120}^{\circ }$ ,AB = BC = CD = AD = 8cm
We know that diagonals of a rhombus are angle bisectors of the vertice angles as well as the diagonals intersect at ${90}^{\circ }$.
So, $\angle OAB={30}^{\circ }$
and $\angle OBA={60}^{\circ }$
Also
$\angle AOB={90}^{\circ }$
In $\Delta$AOB.
$\frac{OA}{AB}=sin{60}^{\circ }$
OA = 8 $\times \frac{\sqrt{3}}{2}=4\sqrt{3}$
AC = OA + OC = $\sqrt{3}+4\sqrt{3}=8\sqrt{3}$
Hence, the longer diagonal 8$\sqrt{3}$cm.