question_answer

A vertical pole AB is standing at the centre B of a square PQRS. If PR subtends an angle of \[{{90}^{o}}\] at the top. A of the pole. then the angle subtended by a side of the square at A is :

A)  \[{{30}^{{}^\circ }}\]                              

B)  \[{{45}^{{}^\circ }}\]

C)  \[{{60}^{{}^\circ }}\]      

D)       None of these

Correct Answer: C

Solution :

  \[PA=PR\] \[\angle APB=\angle ARB={{45}^{o}}\] If \[PR=\sqrt{2x}\,:\] \[PB=\frac{x}{\sqrt{2}}\] From  \[\Delta \,APB,\] \[\tan {{45}^{o}}=\frac{AB}{PB}\Rightarrow AB=PB=\frac{x}{\sqrt{2}}\] \[\therefore \] \[PA=\sqrt{\frac{{{x}^{2}}}{2}+\frac{{{x}^{2}}}{2}}=\sqrt{{{x}^{2}}}=x\] \[\therefore \]    \[QA=PQ=PA=x\] \[\therefore \]    \[\angle PAQ={{60}^{o}}\]