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705-805 (Hard)|   Geometry|      
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Bunuel
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Three semicircles of radius 1 are constructed on diameter AB of a semi 14 Jun 2022, 01:30
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E
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Difficulty:

85% (hard)

Question Stats:

36% (03:09) correct 64% (02:48) wrong based on 22 sessions

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Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?


(A) \(\pi - \sqrt{3}\)

(B) \(\pi - \sqrt{2}\)

(C) \(\frac{\pi + \sqrt{2}}{2}\)

(D) \(\frac{\pi + \sqrt{3}}{2}\)

(E) \(\frac{7\pi}{6} - \frac{\sqrt{3}}{2}\)


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E
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Three semicircles of radius 1 are constructed on diameter AB of a semi 14 Jun 2022, 05:36
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hmm - i tried to solve by using the degree of the smaller circles. not sure
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sonalgmat123
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Three semicircles of radius 1 are constructed on diameter AB of a semi 16 Jun 2022, 08:10
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Hi,

I think it should be E .

Just deduct from the area of bigger semicircle , the area of 3 smaller semi- circle ( which should not include Area of 2 triangles and sectors) .. Complete all the semi circle and then deduct from the the total area of 3 semi-circles the overlapping regions area , which will consists of a triangle and sector on each side .
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Three semicircles of radius 1 are constructed on diameter AB of a semi 16 Jun 2022, 08:26
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My query is how to solve this question in under 2minutes, and that too in exam ?
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Nidzo
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Three semicircles of radius 1 are constructed on diameter AB of a semi 16 Jun 2022, 08:54
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rsama

Important lesson here: Always look at the answer choices.

The area of the shaded region will be: Area of large semi-circle - Area of the region created by the 3 smaller semi-circles

We know that the big semi-circle has a radius of 2, which means it's area is:
\(\frac{πr^2}{2}\)

\(= \frac{π(2)^2}{2}\)

\(= 2π\)

In answer choices A & B the larger semi-circle has an area of π, which means it has a radius of \(\sqrt{2}\). However, we know that the semi-circle has a radius of 2, and that the area is 2π which means that these answer choices have to be wrong.Eliminate both answer choices.

C & D cannot be the area of the shaded area for one simple reason: we subtract the area of the smaller semi-circles from the area of the big semi-circle. We do not add. Eliminate both choices.

Now we may not know how to get to E, or don't want to waste time with solving to get to E, but we know with 100% certainty that the answer cannot be A, B, C or D. So E has to be the answer.

Answer E.
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