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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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20 Nov 2017, 01:11
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98% (00:54) correct 2% (01:43) wrong based on 47 sessions

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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

Attachment:

2017-11-20_1209.png [ 3.42 KiB | Viewed 860 times ]

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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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20 Nov 2017, 01:55
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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20 Nov 2017, 05:29
1
Bunuel wrote:

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

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

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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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20 Nov 2017, 10:48
1
Bunuel wrote:

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

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

$$s^2 - \frac{πs^2}{4}$$

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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20 Nov 2017, 18:36
1
genxer123 wrote:
Bunuel wrote:

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

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

$$s^2 - \frac{πs^2}{4}$$

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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22 Nov 2017, 12:12
Bunuel wrote:

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

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:

s^2 - (1/4)πs^2

s^2 - πs^2/4

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