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

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## 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 $\mathrm{AC}$:$\mathrm{In}\mathrm{right}\mathrm{angle}∆\mathrm{AOD}\phantom{\rule{0ex}{0ex}}\mathrm{sin}30°=\frac{\mathrm{OA}}{\mathrm{AD}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}=\frac{\mathrm{OA}}{20}\mathbf{}\left[\because \mathrm{sin}30°=\frac{1}{2}\right]\phantom{\rule{0ex}{0ex}}⇒\mathrm{OA}=\frac{20}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{OA}=10\mathrm{cm}$Since the diagonal of the rhombus bisect each other,$\therefore AC=2·OA\phantom{\rule{0ex}{0ex}}⇒AC=2×10\phantom{\rule{0ex}{0ex}}⇒AC=20\mathrm{cm}$Step-2: length of the diagonal of $BD$:Find the length of the diagonal $\mathrm{BD}$:$\mathrm{In}\mathrm{right}\mathrm{angle}∆\mathrm{AOD}\phantom{\rule{0ex}{0ex}}\therefore \mathrm{cos}30°=\frac{\mathrm{OD}}{\mathrm{AD}}\phantom{\rule{0ex}{0ex}}⇒\frac{\sqrt{3}}{2}=\frac{\mathrm{OD}}{20}\left[\because \mathrm{cos}30°=\frac{\sqrt{3}}{2}\right]\phantom{\rule{0ex}{0ex}}⇒\mathrm{OD}=\frac{20\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{OD}=10\sqrt{3}\mathrm{cm}$Since the diagonal of the rhombus bisect each other,$\therefore \mathrm{BD}=2·\mathrm{OD}\phantom{\rule{0ex}{0ex}}⇒\mathrm{BD}=2×10\sqrt{3}\phantom{\rule{0ex}{0ex}}⇒\mathrm{BD}=20\sqrt{3}\mathrm{cm}$Hence, the diagonals are $\mathbf{AC}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{}\mathbf{BD}\mathbf{=}\mathbf{20}\sqrt{\mathbf{3}}\mathbf{}\mathbf{cm}$.  Suggest Corrections  10      Similar questions  Explore more