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If the area of each of the three semicircles shown above is 72, and Q

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If the area of each of the three semicircles shown above is 72, and Q  [#permalink]

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New post 28 Nov 2017, 21:13
1
6
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

60% (01:46) correct 40% (02:19) wrong based on 35 sessions

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If the area of each of the three semicircles shown above is 72, and Q is the midpoint of segment PR, what is the area of the shaded region?

(A) 4
(B) 8
(C) 9
(D) 12
(E) 20

Attachment:
2017-11-29_0806.png
2017-11-29_0806.png [ 49.45 KiB | Viewed 729 times ]

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Re: If the area of each of the three semicircles shown above is 72, and Q  [#permalink]

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New post 29 Nov 2017, 19:26
Considering that the Area of each semicircle is 72, the radius = 12/(pi)^1/2.

The key here is noticing that the sector QRS (S being the point of intersection between the Semicircle at an angle and the first semicircle) is the same as the shaded region.

Area (qrs) = (20/360) *pi* r^2

Area = 8

For clarity, if another semicircle is drawn to mimic the semicircle at an angle, it is easy to notice that the area made by the two domes (the result of two semicircles intersecting) will be the same. From this, we can conclude that the shaded area is equal to the area of the sector QRS
Re: If the area of each of the three semicircles shown above is 72, and Q &nbs [#permalink] 29 Nov 2017, 19:26
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