lost in the circles

Geometry Level 2

If ab=bc=bd=dc=ac=11 cm, and the four circles are identical which have a radius 2.5 cm. then

find the area of the ( abdc ).


The answer is 104.5.

Mohamed Hassan by Mohamed Hassan , 25, Egypt

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3 solutions

Swastik Dwibedy
Sep 2, 2015

ABC and BCD are two equilateral triangles. Area of Rhombus = Area of triangle ABC+ Area of triangle BCD= 2 ( Area of Triangle ABC) = 2 {( (3)^1/2 × (11)^2)}/4 =104.786

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Triangle ABC is equilateral.
Draw lines connecting all centers of the inscribed circles. 
(A smaller rhombus will be formed.)
Each side of the smaller rhombus is 2.5 + 2.5 = 5 (s)

Smaller rhombus (abcd)' has area:
2 * (s^2)sqrt(3) / 4 = 25sqrt(3) / 2

(abcd) / (abcd)' = (11/5)^2
(abcd) = (11/5)^2 * 25sqrt(3) / 2 = 104.789

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Herón a b = a c = b c = 11 ab=ac=bc=11 So we can make A = s ( s − a ) ( s − b ) ( s − c ) A=\sqrt{s(s-a)(s-b)(s-c)} Where a , b , and c are the lengths of each side of a triangle, and s is the semiperimeter. And we have to triangles with sides that we know. So 2 A = 2 16.5 ( 5.5 ) 3 = 104.789073858 2A=2\sqrt{16.5(5.5)^{3}}=104.789073858

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