P, Q, S, and R are points on the circumference of a circle of radius r

Question

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 a diameter of the circle. 
What is the Perimeter of the quadrilateral PQSR?

A.  2r(1+\sqrt{3})

B.  2r(2+\sqrt{3})

C.  r(1+\sqrt{5})

D.  2r(1+\sqrt{5})

E.  2r+\sqrt{3}

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Boho · Answer

Sides of quad will be  r, \sqrt{3}r, r, \sqrt{3}r  .
 
A is the correct answer

bv8562 · Answer

Please refer to attached figure.

Diameter PS divides the Quadrilateral PQSR into two SIMILAR RIGHT TRIANGLES

Since, the triangle PQR is an equilateral triangle, therefore angle subtended by each arc: QP = PR = SQ = 120

△QOP is an Isosceles Triangle with sides OP = OQ = r and ∠QOP = 120, therefore, ∠QPO = ∠OQP = 30

△PQS is a Right Triangle with ∠PQS = 90 because the chord PS (Diameter) subtends 180 degree at the center therefore angle subtended in the remaining arc will be half of it.

∴ In △PQS, ∠QPS = 30, ∠PQS = 90 and ∠QSP = 60

We know, ratio of sides in a  30:60:90  =  1: √3 : 2 
∠PQS = 90 = 2r
∠QSP = 60 = √3/2*2r = √3r
∠QPS = 30 = 1/2*2r = r

∴  Perimeter of Quadrilateral PQSR  = 2(√3r) + 2(r) = 2r(√3+1) (A)

Iotaa · Answer

Bunuel

Can you please help with the solution?

ThatDudeKnows · Answer

Like most "hard" geometry questions, you can make your life easier if you get good at estimating and using the answer choices. Let's make r=1. The circumference of the circle is 2pi. Let's call that 6.3.

How much shorter is QS than the arcQS? BAREly.
How much shorter is RS than the arcRS? BAREly.
How much shorter is PR than the arcPR? Okay, maybe a little, but it's definitely a decent amount more than half of arcPR.
How much shorter is PQ than the arcPQ? Okay, maybe a little, but it's definitely a decent amount more than half of arcPQ.

Look at the answer choices:
A.  2r(1+\sqrt{3})  = 2(2.7) = 5.4 . . . Keep it.

B.  2r(2+\sqrt{3})  = 2(3.7) . . . That's bigger than the circle. Nope.

C.  r(1+\sqrt{5})  = 3.23 . . . That's barely more than half the circle. Nope.

D.  2r(1+\sqrt{5})  = 2(3.23) . . . That's bigger than the circle. Nope.

E.  2r+\sqrt{3}  = 2+1.7 = 3.7 . . . That's less than 60% of the circle. Nope.

Answer choice A.

FWIW, It's worthwhile to know the following:
 \sqrt{2} = 1.4 
 \sqrt{3} = 1.7 
 \sqrt{5} = 2.3

P, Q, S, and R are points on the circumference of a circle of radius r

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GMAT Problem Solving (PS) Questions

P, Q, S, and R are points on the circumference of a circle of radius r

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