question_answer

     An equilateral triangle and a regular hexagon are inscribed in a given circle. If a and b are the lengths of their sides respectively, then which one of following is correct?                       

A)  \[{{a}^{2}}=2{{b}^{2}}\]               

B)  \[{{b}^{2}}=3{{a}^{2}}\]

C)  \[{{b}^{2}}=2{{a}^{2}}\]               

D)  \[{{a}^{2}}=3{{b}^{2}}\]

Correct Answer: D

Solution :

We know altitude of equilateral             \[\Delta ABC\]is \[\frac{\sqrt{3}}{2}a.\] \[\therefore \]Length of \[OC=\frac{\sqrt{3}}{2}a\times \frac{2}{3}=\frac{a}{\sqrt{3}}=\]radius Also,     \[DF=b\] \[\Rightarrow \]   \[DE=\frac{b}{2}\] In \[\Delta ODE,\]\[\cos 60{}^\circ =\frac{DE}{OD}=\frac{b/2}{a/\sqrt{3}}\] \[\Rightarrow \]   \[\frac{1}{2}=\frac{\sqrt{3}b}{2a}\] \[\Rightarrow \]   \[a=\sqrt{3}b\]\[\Rightarrow \]\[{{a}^{2}}=3{{b}^{2}}\]