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P, Q, R, S four points lie in a plane and PQ = PR = QR = PS then how many possible values of angle QSR can exist?

I think 2 values because PQRS is either a square or rhombus.

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As written, triangle PQR is equilateral... –  The Chaz 2.0 Jul 7 '12 at 15:37
    
There are 2 values, but PQRS is neither a square nor a rhombus, since, for example, PQR is an equilateral triangle. See my answer below. –  Omar Antolín-Camarena Jul 7 '12 at 15:46

1 Answer 1

up vote 3 down vote accepted

Let $r = PQ = PR = QR = PS$. The triangle $PQR$ is equilateral with side length $r$ and $S$ is some point on the circle with center $P$ and radius $r$ (this circle also passes through $Q$ and $R$). The angle $QSR$ is therefore half of $QPR$, i.e., $30$ degrees, when $S$ is on the big arc $QR$ of the circle, and is $180$ minus that, i.e., $150$ degrees, when $S$ is on the small arc $QR$ of the circle.

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You might want to mention the Inscribed Angle Theorem and use this diagram. –  robjohn Jul 7 '12 at 16:51
    
Thanks for the diagram, @robjohn. I didn't refer to the inscribed angle theorem by name, but I did include a link to MathWorld's page on it. –  Omar Antolín-Camarena Jul 7 '12 at 18:56

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