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The vertex of the square MNOP is located at the center of circle O. If

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The vertex of the square MNOP is located at the center of circle O. If [#permalink]

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New post 05 Feb 2018, 05:41
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The vertex of the square MNOP is located at the center of circle O. If arc NP is \(4\pi\) units long, then which of the following is the perimeter of the square MNOP ?

(A) 32

(B) \(32\pi\)

(C) 64

(D) \(64\pi\)

(E) \(72\pi\)

[Reveal] Spoiler:
Attachment:
2018-02-05_1739.png
2018-02-05_1739.png [ 10.3 KiB | Viewed 377 times ]
[Reveal] Spoiler: OA

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Re: The vertex of the square MNOP is located at the center of circle O. If [#permalink]

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New post 05 Feb 2018, 05:51
Bunuel wrote:
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The vertex of the square MNOP is located at the center of circle O. If arc NP is \(4\pi\) units long, then which of the following is the perimeter of the square MNOP ?

(A) 32

(B) \(32\pi\)

(C) 64

(D) \(64\pi\)

(E) \(72\pi\)

[Reveal] Spoiler:
Attachment:
2018-02-05_1739.png


Arc length = \(2\pi\)r(center angle/360)

Here \(4\pi\)=\(2\pi\)r(90/360)

r=8

Perimeter =32

Hence A
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Re: The vertex of the square MNOP is located at the center of circle O. If [#permalink]

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New post 05 Feb 2018, 12:12
Bunuel wrote:
Image
The vertex of the square MNOP is located at the center of circle O. If arc NP is \(4\pi\) units long, then which of the following is the perimeter of the square MNOP ?

(A) 32

(B) \(32\pi\)

(C) 64

(D) \(64\pi\)

(E) \(72\pi\)

[Reveal] Spoiler:
Attachment:
The attachment 2018-02-05_1739.png is no longer available

Attachment:
2018-02-05_1739.ed.png
2018-02-05_1739.ed.png [ 20.89 KiB | Viewed 164 times ]

Length of arc NP = \(4π\)
Arc length is a fraction of circumference.
Derive that fraction from central angle/circle angle.
From circumference, derive radius, which equals side length of the square.

• Arc length = fraction of circumference

To find that fraction, we need the central angle of sector NOP
A sector's arc length is the portion of the circumference that is subtended by sector's central angle.

• Central angle of sector NOP?

Square MNOP has a vertex located at center O
A square's vertex = 90°
Hence the central angle of sector NOP = 90°

• Sector NOP as a fraction of the circle?

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

•Circumference of circle?

Sector NOP's arc = \(\frac{1}{4}\) of circumference
Length of arc NP = 4π
4π = \(\frac{1}{4}\) circumference
Circumference = 16π

• Radius? Derive radius from circumference

Circumference = \(2πr = 16π\)
\(r = 8\)
\(r = s\)
, side of square
\(s = 8\)

• Perimeter of square? \(= 4s\)
\(4s = 4 * 8 = 32\)

Answer A
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Re: The vertex of the square MNOP is located at the center of circle O. If [#permalink]

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New post 06 Feb 2018, 17:03
Bunuel wrote:
Image
The vertex of the square MNOP is located at the center of circle O. If arc NP is \(4\pi\) units long, then which of the following is the perimeter of the square MNOP ?

(A) 32

(B) \(32\pi\)

(C) 64

(D) \(64\pi\)

(E) \(72\pi\)

[Reveal] Spoiler:
Attachment:
2018-02-05_1739.png



Since arc NP is 4π, and 4π corresponds to a 90 degree central angle, then 4π is 90/360 = 1/4 of the circumference of the circle.

Thus, the circumference is 4π x 4 = 16π. Thus, the diameter is 16, and the radius is 8.

So, side NO = side of square = radius = 8, and the perimeter of the square is 8 x 4 = 32.

Answer: A
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Re: The vertex of the square MNOP is located at the center of circle O. If   [#permalink] 06 Feb 2018, 17:03
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