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05 Feb 2018, 06:41
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A
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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) \(72\pi\)
Attachment:
2018-02-05_1739.png [ 10.3 KiB | Viewed 7353 times ]
05 Feb 2018, 06:51
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Bunuel
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) \(72\pi\)
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
05 Feb 2018, 13:12
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Bunuel
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) \(72\pi\)
Attachment:
The attachment 2018-02-05_1739.png is no longer available
Attachment:
2018-02-05_1739.ed.png [ 20.89 KiB | Viewed 6347 times ]
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
