geometry STD X

P, Q, S and R are points on the circumference of a circle of radius  "r". such that PQR is an equilateral triangle and PS is the diameter of the circle. What is the perimeter of the quadrilateral PQRS?

   






















Consider triangle POR.
we have cos 30 = (a/2)/r
or a = √3r
Since PS is the diameter, triangle PRS is right angled
so RS² = PS² – PR²

or RS² = (2r)² – (√3r)²

or, RS = r

So, perimeter of PQSR = 2 x (r + √3r) = 2r (1 + √3)