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Updated on: 24 Jul 2019, 23:13
Originally posted by CareerGeek on 24 Jul 2019, 10:03.
Last edited by CareerGeek on 24 Jul 2019, 23:13, edited 2 times in total.
Last edited by CareerGeek on 24 Jul 2019, 23:13, edited 2 times in total.
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
59% (02:26) correct
41%
(02:19)
wrong
based on 41
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Find the ratio of area of shaded circle to the area of Equilateral triangle
A. π : 36
B. 2π : 27
C. π : 54
D. π : 27\(\sqrt{3}\)
E. π : 108
Attachment:
1.png
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Dillesh4096
Find the ratio of area of shaded circle to the area of Equilateral triangle
A. π : 36
B. 2π : 27
C. π : 54
D. π : 27
E. π : 108
Attachment:
1.png
For an eq \(\triangle\) of side X, area is \(\sqrt{3}X^2/4\)
For the bigger circle, radius is \(\sqrt{3}X/6\) as centroid and incenter coincides and centroid divides median in the ratio 2:1.
This also means that if a line, parallel to the base of the \(\triangle\), is drawn tangentially to both circles, we would have a bigger \(\triangle\) of side X with 2 circles and a smaller \(\triangle\) of side X/3 with the one smaller circle.
The ratio of sides for the smaller and bigger \(\triangle\) - 1:3. Also as the smaller \(\triangle\) is similar to the bigger \(\triangle\) and thus equilateral \(\triangle\), the radius of the smaller circle is \(\sqrt{3}X/18\)
Therefore ratio of areas would be - \((\pi*\sqrt{3}X/18*\sqrt{3}X/18)\)/\(\sqrt{3}X^2/4\) = \(\pi*\sqrt{3}/81\) or its equivalent.
Experts, please suggest how is E - \(\pi/108\) correct and help identify my mistake, if wrong.
24 Jul 2019, 23:14
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aesop
Dillesh4096
Find the ratio of area of shaded circle to the area of Equilateral triangle
A. π : 36
B. 2π : 27
C. π : 54
D. π : 27
E. π : 108
Attachment:
1.png
For an eq \(\triangle\) of side X, area is \(\sqrt{3}X^2/4\)
For the bigger circle, radius is \(\sqrt{3}X/6\) as centroid and incenter coincides and centroid divides median in the ratio 2:1.
This also means that if a line, parallel to the base of the \(\triangle\), is drawn tangentially to both circles, we would have a bigger \(\triangle\) of side X with 2 circles and a smaller \(\triangle\) of side X/3 with the one smaller circle.
The ratio of sides for the smaller and bigger \(\triangle\) - 1:3. Also as the smaller is similar to the bigger and thus equilateral \(\triangle\), the radius of the smaller circle is \(\sqrt{3}X/18\)
Therefore ratio of areas would be - \((\pi*\sqrt{3}X/18*\sqrt{3}X/18)\)/\(\sqrt{3}X^2/4\) = \(\pi*\sqrt{3}/81\) or its equivalent.
Experts, please suggest how is E - \(\pi/108\) correct and help identify my mistake, if wrong.
Hi aesop,
You answer is correct.
OA is edited. (D)
Thanks
