Consider an

rectangular glassboard \(ABCD\) , with surfaces \(AB\), \(BC\), \(CD\) areperfectly silvered. An smalllaser lightis set in the corner \(A\) (negligible dimensions) . Now just for fun , this laser light is set in such a way so that light ray first travel from A making an angle \(\theta \) with side \(AD\) and thentotally reflectedby surfaces \(DC\) , then \(CB\) , then \(BA\) and finallyrefractedout from surface \(AD\). If during this event , anquadrilateraland antriangleare traced out as shown & their point of intersection is \(P\), Such thatAreaof triangle( \(APT\) ) is equal to that of quadrilateral( \(PQRS\) ).

**Given that** :

\(AB=CD=2(BC)=2(AD)\)

\(\text{Laws of reflection holds perfectly}\).

Then find that particular angle \(\theta \) in degrees or just simply prove that \(\tan { \theta } =\cfrac { a-\sqrt { b } }{ c } \) and find \(a+b+c\) , for which such event is possible.

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