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In the figure above, XYZW is a square with sides of length s. If YW is

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In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 20 Nov 2017, 00:11
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In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2
(B) s^2 – πs^2
(C) s^2 – πs^2/4
(D) 4s – πs
(E) 4s – πs/4

[Reveal] Spoiler:
Attachment:
2017-11-20_1209.png
2017-11-20_1209.png [ 3.42 KiB | Viewed 654 times ]
[Reveal] Spoiler: OA

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Re: In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 20 Nov 2017, 00:55
I think the answer is C.

Calculated by making the provided diagram into a square w/ circle inscribed (ie. x4). [(Area of square) - (area of circle)]/4.

If there's a faster way to solve this, I would love to know how!

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Re: In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 20 Nov 2017, 04:29
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Bunuel wrote:
Image
In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2
(B) s^2 – πs^2
(C) s^2 – πs^2/4
(D) 4s – πs
(E) 4s – πs/4

[Reveal] Spoiler:
Attachment:
2017-11-20_1209.png



Area of shaded region = Area of Square - Area of a quarter of a circle

Length of side of the Square = s
Radius of the circle = s

Area of shaded region = s^2 - (1/4)*π*s^2


ANswer: Option C
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In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 20 Nov 2017, 09:48
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Bunuel wrote:
Image
In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2
(B) s^2 – πs^2
(C) s^2 – πs^2/4
(D) 4s – πs
(E) 4s – πs/4

[Reveal] Spoiler:
Attachment:
2017-11-20_1209.png

yukidaruma , I think this way might be faster (and it is essentially what you figured out, condensed).

(Area of square) - (Area of sector WXY) = area of shaded region

In these kinds of problems, almost always, the key is: "The sector is what fraction of the circle?"

Find that fraction, in this case, by using the sector's central angle and the 360° of a circle.

The central angle of this sector is the vertex of what we are told is a square. So the sector's central angle = 90°. Thus:

\(\frac{Part}{Whole}=\frac{SectorArea}{CircleArea}=\frac{90}{360}=\frac{1}{4}\)

The sector is \(\frac{1}{4}\) of the circle.* We need the circle's area divided by 4.

The circle with radius = \(s\) has area: \(πr^2 = πs^2\)
Sector area? \(\frac{πs^2}{4}\)

Area of square = \(s^2\)

(Area of square) - (Area of sector) = area of shaded region

\(s^2 - \frac{πs^2}{4}\)

Answer C

Hope it helps.

*This fraction can be used in a few ways. Example: to find arc length, which here would = 1/4 of circle's circumference; or the reverse, to find circumference from arc length (here, = arc length * 4).

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Re: In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 20 Nov 2017, 17:36
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genxer123 wrote:
Bunuel wrote:
Image
In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2
(B) s^2 – πs^2
(C) s^2 – πs^2/4
(D) 4s – πs
(E) 4s – πs/4

[Reveal] Spoiler:
Attachment:
2017-11-20_1209.png

yukidaruma , I think this way might be faster (and it is essentially what you figured out, condensed).

(Area of square) - (Area of sector WXY) = area of shaded region

In these kinds of problems, almost always, the key is: "The sector is what fraction of the circle?"

Find that fraction, in this case, by using the sector's central angle and the 360° of a circle.

The central angle of this sector is the vertex of what we are told is a square. So the sector's central angle = 90°. Thus:

\(\frac{Part}{Whole}=\frac{SectorArea}{CircleArea}=\frac{90}{360}=\frac{1}{4}\)

The sector is \(\frac{1}{4}\) of the circle.* We need the circle's area divided by 4.

The circle with radius = \(s\) has area: \(πr^2 = πs^2\)
Sector area? \(\frac{πs^2}{4}\)

Area of square = \(s^2\)

(Area of square) - (Area of sector) = area of shaded region

\(s^2 - \frac{πs^2}{4}\)

Answer C

Hope it helps.

*This fraction can be used in a few ways. Example: to find arc length, which here would = 1/4 of circle's circumference; or the reverse, to find circumference from arc length (here, = arc length * 4).


genxer123 thanks for the tip!

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Re: In the figure above, XYZW is a square with sides of length s. If YW is [#permalink]

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New post 22 Nov 2017, 11:12
Bunuel wrote:
Image
In the figure above, XYZW is a square with sides of length s. If YW is the arc of a circle with center X, which of the following is the area of the shaded region in terms of s?

(A) πs^2 – (s/2)^2
(B) s^2 – πs^2
(C) s^2 – πs^2/4
(D) 4s – πs
(E) 4s – πs/4

[Reveal] Spoiler:
Attachment:
2017-11-20_1209.png



We see that a side of the square = radius of the circle = s

Since the area of the square = s^2 and the area of ¼ of the circle = (1/4)πs^2, we have:

Area of shaded region:

s^2 - (1/4)πs^2

s^2 - πs^2/4

Answer: C
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Kudos [?]: 991 [0], given: 5

Re: In the figure above, XYZW is a square with sides of length s. If YW is   [#permalink] 22 Nov 2017, 11:12
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