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Sub 505 (Easy)|   Geometry|      
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Bunuel
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In the figure above, XYZW is a square with sides of length s. If YW is 20 Nov 2017, 01:11
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C
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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

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C
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yukidaruma
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In the figure above, XYZW is a square with sides of length s. If YW is 20 Nov 2017, 01:55
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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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In the figure above, XYZW is a square with sides of length s. If YW is 20 Nov 2017, 05:29
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Bunuel

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

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2017-11-20_1209.png
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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 20 Nov 2017, 10:48
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Bunuel

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

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Attachment:
2017-11-20_1209.png
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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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In the figure above, XYZW is a square with sides of length s. If YW is 20 Nov 2017, 18:36
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genxer123
Bunuel

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

Show 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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genxer123 thanks for the tip!
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In the figure above, XYZW is a square with sides of length s. If YW is 22 Nov 2017, 12:12
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Bunuel

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

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Attachment:
2017-11-20_1209.png
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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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