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CAT 2005 — QA Question 24

Basics of CirclesEasy

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?

Answer & solution

Correct answer: 2r(1 + 3 )

  • 2r(1 + 3)

  • B

    2r(2 + 3)

  • C

    r(1 + 5)

  • D

    2r + 3

Solution

∆ PQR is an equilateral triangle and PS is the diameter.

∴ m ∠PQS = m ∠PRS = 90°   (angles subtended in a semi-circle)

and m ∠PQM = m ∠PRM = m ∠QPR = 60°   (each angle in an equilateral triangle = 60°)

PS bisects ∠QPS as it is the median of ∆PQR.

m ∠PMQ = m ∠PMR = 90°

∴ m ∠QPS = m ∠RPS = 30°

∴ m ∠PSQ = m ∠PSR = 60°

Radius = r

∴ PS = 2r

As ∆ PQS, ∆ PQM, ∆ MQS are 30°-60°-90° triangles,

QS = r, PQ = √3r

Similarly, RS = r and PR = √3r

∴ Perimeter of quadrilateral PQRS = 2r + 2√3r = 2r(1 + √3)

Hence, option (a).

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CAT 2005 QA Q24: P, Q, S, and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triang — Solution | TheCATExam