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CBSE Class 10 Answered

Three equal circles are placed inside an equilateral triangle such that any circle is tangential to two sides of the equilateral triangle and to two other circles. What is the ratio of the areas of one circle to that of the triangle
Asked by rushabhjain.a | 07 Feb, 2019, 10:51: AM
answered-by-expert Expert Answer
Figure shows the equilateral ΔABC that enclosed the three circles as described in the question.
 
let us divide side AB as AF, FD and DB. From F a tangent is drawn so that this tangent meets the circle with centre as O at G.
 
Since BD and BE are tangents drawn from B to circle and begin mathsize 12px style angle end styleB is 60°, it can be shown that ΔBDO  is a right-angled triangle
and other angles are 30° and 60°. Hence it can be worked out that if r is radius of circle then BD = √3r
 
Similarly AF = DF = √3r
 
Hence side AB = 3√3r
 
Ratio of area of circle to that of triangle = begin mathsize 12px style fraction numerator straight pi space straight r squared over denominator begin display style fraction numerator square root of 3 over denominator 4 end fraction open parentheses 3 square root of 3 space r close parentheses squared end style end fraction space equals space fraction numerator 4 straight pi over denominator 27 square root of 3 end fraction end style
Answered by Thiyagarajan K | 08 Feb, 2019, 10:58: AM

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