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In the figure above, ABCD is a square, and P and Q are the centers of

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In the figure above, ABCD is a square, and P and Q are the centers of  [#permalink]

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New post 07 Oct 2016, 02:37
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

62% (01:47) correct 38% (01:46) wrong based on 177 sessions

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In the figure above, ABCD is a square, and P and Q are the centers of identical semicircles BF and AE, respectively. If DE=EA=AF=FB=1, then what is the perimeter of the shaded region?

A. 4+π/2
B. 6
C. 8
D. 6+π
E. 6+2π

Attachment:
T6357.png
T6357.png [ 5.24 KiB | Viewed 2319 times ]

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In the figure above, ABCD is a square, and P and Q are the centers of  [#permalink]

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New post 07 Oct 2016, 05:14
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1
Given data
DE=EA=AF=FB=1

Radius of the semicircles are 1/2.
Circumference of the semicircle = pi * radius = pi/2
Also side of the square is 2(DE+EA).

Perimeter = 2+2+1(DE)+1(AF)+\(\frac{\pi}{2}\)(perimeter of semi-circle1)+\(\frac{\pi}{2}\)(perimeter of semi-circle2) = \(6 + \pi\)

Therefore, perimeter of the shaded region is \(6 + \pi\) (Option D)
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Re: In the figure above, ABCD is a square, and P and Q are the centers of  [#permalink]

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New post 17 Feb 2018, 11:51
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Re: In the figure above, ABCD is a square, and P and Q are the centers of  [#permalink]

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New post 23 Oct 2018, 16:22
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Top Contributor
Bunuel wrote:
Image
In the figure above, ABCD is a square, and P and Q are the centers of identical semicircles BF and AE, respectively. If DE=EA=AF=FB=1, then what is the perimeter of the shaded region?

A. 4+π/2
B. 6
C. 8
D. 6+π
E. 6+2π

Attachment:
T6357.png


Add given info to diagram:
Image

We can see that each side of the square has length 2
Image

At this point, we can see that the perimeter consists of some STRAIGHT parts and some CURVED parts
Image

Since the curved parts are SEMICIRCLES, we can use the circumference formula, which says: circumference = (diameter)(π)
Since a semiCIRCLE is half of a circle, we can say: perimeter of a SEMIcircle = (1/2)(diameter)(π)

Since each semicircle has diameter 1, the perimeter of 1 semicircle = (1/2)(1)(π) = π/2
Image

So, the perimeter of the given figure = 2 + 2 + 1 + 1 + π/2 + π/2
= 6 + π

Answer: D

Cheers,
Brent
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Re: In the figure above, ABCD is a square, and P and Q are the centers of   [#permalink] 23 Oct 2018, 16:22
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