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

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Math Expert
Joined: 02 Sep 2009
Posts: 50078
If the area of each of the three semicircles shown above is 72, and Q  [#permalink]

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28 Nov 2017, 21:13
1
6
00:00

Difficulty:

45% (medium)

Question Stats:

50% (01:44) correct 50% (02:15) 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 [ 49.45 KiB | Viewed 844 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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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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# If the area of each of the three semicircles shown above is 72, and Q

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