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# In the figure given below, ABC and CDE are two identical semi-circles

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Math Expert
Joined: 02 Sep 2009
Posts: 54369
In the figure given below, ABC and CDE are two identical semi-circles  [#permalink]

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29 Jan 2016, 03:23
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Difficulty:

45% (medium)

Question Stats:

68% (02:11) correct 32% (02:28) wrong based on 117 sessions

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In the figure given below, ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region?

A. 4π - 1
B. 3π - 1
C. 2π - 4
D. ½(3π - 1)
E. 2π - 2

Attachment:

Untitled.png [ 25.32 KiB | Viewed 2980 times ]

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Posts: 7556
Re: In the figure given below, ABC and CDE are two identical semi-circles  [#permalink]

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29 Jan 2016, 09:12
1
Bunuel wrote:
In the figure given below, ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region?

A. 4π - 1
B. 3π - 1
C. 2π - 4
D. ½(3π - 1)
E. 2π - 2

Attachment:
Untitled.png

Hi,
the figure may look complicated but can be actually taken as one circle divided into two half..
We can work on one semicircle and then multiply the area of shaded region by 2..

join point B with th center of the diameter say, O..
so OBC is a right angle triangle, as B is the midpoint of arc AC...
what will be the area of shaded region..
=area of sector BC - area of triangle OBC..

sector BC is 1/4 of the circle of radius 2, so area = π *2^2/4= π ..
area of OBC= 1/2* (OB*OC)= 2^2/2=2..
so area of one shaded region= π - 2..
area of total shaded region= 2(π - 2)
C
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In the figure given below, ABC and CDE are two identical semi-circles  [#permalink]

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29 Jan 2016, 09:54
1
I took a similar approach to Chetan4u, but worked with a square inside the circle instead of a triangle.

If you inscribe a square inside a circle, then there will be 4 small regions inside the circle but outside the square. The question is asking us to find the area of two of these regions. Pictures are so much easier:

Attachment:

Square in circle.png [ 4.56 KiB | Viewed 2438 times ]

Now, the area of the circle is $$\pi r^2 = 4\pi$$
The length of one side of the square is $$\sqrt{2^2+2^2} = \sqrt{8}$$
So the area of the square is $$(\sqrt{8})^2 = 8$$
And the area of the shaded region is half the area outside the square
$$\frac{1}{2}(4\pi-8) = 2\pi-4$$

*For some reason the math editor is adding a vertical line at the end of each expression... it is not part of the expression and should be ignored.
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Re: In the figure given below, ABC and CDE are two identical semi-circles  [#permalink]

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29 Jan 2016, 10:22
1
Bunuel wrote:
In the figure given below, ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region?

A. 4π - 1
B. 3π - 1
C. 2π - 4
D. ½(3π - 1)
E. 2π - 2

Attachment:
The attachment Untitled.png is no longer available

Another method:

In GMAT PS, all figures are drawn to scale unless otherwise noted. So will use the figure given. Carefully note that each of the 2 shaded areas is approx. 1/6th the area of the corresponding semi circle as shown below:

Attachment:

1-29-16 12-21-16 PM.jpg [ 17.35 KiB | Viewed 2428 times ]

Thus, the shaded area total = $$2*(1/6)*\pi*4/2 \approx 2$$sq. units. (with $$\pi \approx 3$$)

Now analyse the options and see which one gives a value closest to 2 sq. units.

A. 4π - 1 $$\approx$$ 11. Eliminate
B. 3π - 1 $$\approx$$ 8. Eliminate
C. 2π - 4 $$\approx$$ 2. Keep
D. ½(3π - 1) $$\approx$$ 4. Eliminate
E. 2π - 2 $$\approx$$ 4. Eliminate

Thus C is the correct answer.
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Re: In the figure given below, ABC and CDE are two identical semi-circles  [#permalink]

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25 Aug 2017, 16:02
Is there an easy way to do this?
Re: In the figure given below, ABC and CDE are two identical semi-circles   [#permalink] 25 Aug 2017, 16:02
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