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A circle with radius 6 cm is inscribed inside an equilateral triangle ABC. Three smaller circle are drawn touching the incircle and the sides of ABC as shown in the figure. Another triangle is formed by joining centres P,Q and R of these smaller circles. What is the perimeter of triangle PQR?
  • a)
    24√3
  • b)
    36√3
  • c)
    8√3
  • d)
    12√3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A circle with radius 6 cm is inscribed inside an equilateral triangle ...

Consider radius of each smaller circle be r and that of the larger circle be R = 6 cm.
Construct PD such that PD is perpendicular to AB.
In right triangle ADP, AP = DP/cos(APD) = DP/ cos(60) = 2DP = 2r
In right triangle OFA, OF/cos(AOF) = AO ⇒ R/(1/2) = AP+OP  ⇒ 2R=2r+r+R   ⇒ R = 3r
 In right triangle OEP, EP=OPcos(EPO) = (R + r)((√3)/2 = (4R/3)((√3) = 2R/(√3)
PQ=2EP = 4R/√3)
Perimeter = 12R/(√3) = 24√3 cm (R = 6cm)
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Most Upvoted Answer
A circle with radius 6 cm is inscribed inside an equilateral triangle ...
Since, ∆ABC is equilateral,

Bigger circle's radius = 2 * smaller circle's radius

So, smaller circle's radius = 3cm.


From the fig., Angle PQR = 60 degrees

sin 60 = height of ∆PQR/ hypotenuse of ∆PQR

√3/2 = (3+6+3) /PQ

PQ = 24 / √3

Perimeter of ∆PQR = 3 PQ = 3 * 24/√3 = 24√3.

Hence, option A is correct.
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