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Two identical circles of area 144π are shown above. If the distance

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Two identical circles of area 144π are shown above. If the distance [#permalink]

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New post 27 Jan 2018, 07:42
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Two identical circles of area 144π are shown above. If the distance from point A to point B is 12, what is the perimeter of the shaded region?

A. 4π
B. 6π
C. 8π
D. 12π
E. 24π

[Reveal] Spoiler:
Attachment:
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image008.jpg [ 2.48 KiB | Viewed 560 times ]
[Reveal] Spoiler: OA

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Re: Two identical circles of area 144π are shown above. If the distance [#permalink]

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New post 27 Jan 2018, 08:42
C.

Let the centre of the two triangles be O and P
Area=144π. r^2=144.
Solving for area we get the radius of each circle as 12. Also AB=12. Hence triangle OAB and PAB are equilateral triangles with side 12.

Hence Angle AOB=APB=60deg. Hence arc length of AB=2πr*60/360 = 2π*12/6=4π. Hence two arc length=2*4π =8π
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Re: Two identical circles of area 144π are shown above. If the distance [#permalink]

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New post 27 Jan 2018, 09:52
It says area of two identical circles to be 144pie nt one circle alone.


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Two identical circles of area 144π are shown above. If the distance [#permalink]

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New post 27 Jan 2018, 12:24
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Bunuel wrote:
Image
Two identical circles of area 144π are shown above. If the distance from point A to point B is 12, what is the perimeter of the shaded region?

A. 4π
B. 6π
C. 8π
D. 12π
E. 24π

[Reveal] Spoiler:
Attachment:
The attachment image008.jpg is no longer available

Attachment:
image008ed.png
image008ed.png [ 32.15 KiB | Viewed 349 times ]

Perimeter of the shaded region = arc length AB * 2 (i.e., not chord AB)
Identical circles have identical circumferences
Arc length = fraction of circumference

To find that fraction, we need the central angle for the sector whose arc is AB
(A sector's arc length is that portion of the circumference subtended by sector's central angle.)

1) Sector with arc AB? Draw a triangle in one of the circles
Draw two lines, from center O to A, and from O to B. OA and OB are radii

2) Side lengths OA and OB of ∆ OAB = radius length
Derive radius from area: \(144π = πr^2\)
\(r^2 = 144\), \(r = 12\)


3) ∆ OAB is equilateral; all sides have equal length
AB = OA = OB = 12, hence
All angles of ∆ OAB = 60°
Central angle of of sector AOB = 60°

3) Sector AOB as a fraction of the circle?

\(\frac{Part}{Whole}=\frac{SectorAngle}{360}=\frac{60}{360}=\frac{1}{6}\)

4) Length of arc AB? \(\frac{1}{6}\) of circumference
Circle's circumference, r = 12: \(= 2πr = (2π * 12) = 24π\)
Arc Length: \((\frac{1}{6} * 24π) = 4π\)

5) Perimeter of shaded region: (2 * length of arc AB) = \((2 * 4π) = 8π\)

Answer C
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Two identical circles of area 144π are shown above. If the distance   [#permalink] 27 Jan 2018, 12:24
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Two identical circles of area 144π are shown above. If the distance

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