CAT 2005 — QA Question 24
Answer the next 2 questions based on the information given below.
Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.
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
2r(1 + )
- B
2r(2 + )
- C
r(1 + )
- D
2r +
â 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 âPQR.
m ∠PMQ = m ∠PMR = 90°
∴ m ∠QPS = m ∠RPS = 30°
∴ m ∠PSQ = m ∠PSR = 60°
Radius = r
∴ PS = 2r
As â PQS, â PQM, â 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).