A) \[2\,(2+\sqrt{5})\]
B) \[\sqrt{3}\,(2+\sqrt{5})\]
C) \[\frac{\sqrt{5}+2}{2\sqrt{2}}\]
D) \[\frac{2+\sqrt{5}}{\sqrt{3}}\]
Correct Answer: B
Solution :
In \[\Delta ABC,\]\[\angle A=30{}^\circ \], \[BC=2+\sqrt{5}\] |
\[\frac{a}{\sin A}=2R\]\[\Rightarrow \]\[\frac{2+\sqrt{5}}{\sin 30{}^\circ }=2R\] |
\[R=\frac{2+\sqrt{5}}{2\times \frac{1}{2}}=2+\sqrt{5}\] |
Now, \[AH=2R\cos A\] |
\[\because \]\[AH=2\,(2+\sqrt{5})\cos 30{}^\circ \]\[\Rightarrow \]\[AH=2\,(2+\sqrt{5})\frac{\sqrt{3}}{2}=(2+\sqrt{5})(\sqrt{3})\] |
\[\therefore \]\[AH=\sqrt{3}\,(2+\sqrt{5})\] |
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