0
$\begingroup$

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.

  • 0
    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.2012-07-07

1 Answers 1

3

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.

  • 0
    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.2012-07-07