Question

A rhombus of side $20\mathrm{cm}$ has two angles of ${60}^{\circ }$ each .Find the length of the diagonal.

Answer 1

Step-1: length of the diagonal of $AC$:

The diagonals of a rhombus bisect each other at a right angle.

$\therefore \angle \mathrm{AOD}=90°$

Find the length of diagonal $\mathrm{AC}$:

$$\begin{gathered} \mathrm{In}\mathrm{right}\mathrm{angle}∆\mathrm{AOD} \\ \sin 30°=\frac{\mathrm{OA}}{\mathrm{AD}} \\ \Rightarrow \frac{1}{2}=\frac{\mathrm{OA}}{20}\mathbf{}\left[\because \sin 30°=\frac{1}{2}\right] \\ \Rightarrow \mathrm{OA}=\frac{20}{2} \\ \Rightarrow \mathrm{OA}=10\mathrm{cm} \end{gathered}$$

Since the diagonal of the rhombus bisect each other,

$$\begin{gathered} \therefore AC=2·OA \\ \Rightarrow AC=2\times 10 \\ \Rightarrow AC=20\mathrm{cm} \end{gathered}$$

Step-2: length of the diagonal of $BD$:

Find the length of the diagonal $\mathrm{BD}$:

$$\begin{gathered} \mathrm{In}\mathrm{right}\mathrm{angle}∆\mathrm{AOD} \\ \therefore \cos 30°=\frac{\mathrm{OD}}{\mathrm{AD}} \\ \Rightarrow \frac{\sqrt{3}}{2}=\frac{\mathrm{OD}}{20}\left[\because \cos 30°=\frac{\sqrt{3}}{2}\right] \\ \Rightarrow \mathrm{OD}=\frac{20\sqrt{3}}{2} \\ \Rightarrow \mathrm{OD}=10\sqrt{3}\mathrm{cm} \end{gathered}$$

Since the diagonal of the rhombus bisect each other,

$$\begin{gathered} \therefore \mathrm{BD}=2·\mathrm{OD} \\ \Rightarrow \mathrm{BD}=2\times 10\sqrt{3} \\ \Rightarrow \mathrm{BD}=20\sqrt{3}\mathrm{cm} \end{gathered}$$

Hence, the diagonals are $\mathbf{AC}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{}\mathbf{BD}\mathbf{=}\mathbf{20}\sqrt{\mathbf{3}}\mathbf{}\mathbf{cm}$.