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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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05 Feb 2018, 05:41
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Difficulty:

15% (low)

Question Stats:

85% (01:14) correct 15% (01:35) wrong based on 52 sessions

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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 858 times ]

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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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05 Feb 2018, 05:51
Bunuel wrote:

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
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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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05 Feb 2018, 12:12
Bunuel wrote:

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 603 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

Circumference = $$2πr = 16π$$
$$r = 8$$
$$r = s$$
, side of square
$$s = 8$$

• Perimeter of square? $$= 4s$$
$$4s = 4 * 8 = 32$$

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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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06 Feb 2018, 17:03
Bunuel wrote:

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

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.

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

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