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# In the figure above, AC is the diameter of a circle with center O. If

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Joined: 02 Sep 2009
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In the figure above, AC is the diameter of a circle with center O. If [#permalink]

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11 Apr 2018, 21:27
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In the figure above, AC is the diameter of a circle with center O. If the area of the circle is 81π, what is the length of minor arc BC?

A. 3.5π
B. 6π
C. 7π
D. 8.5π
E. 9π

[Reveal] Spoiler:
Attachment:

Arc_Length.png [ 8.96 KiB | Viewed 387 times ]
[Reveal] Spoiler: OA

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Re: In the figure above, AC is the diameter of a circle with center O. If [#permalink]

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12 Apr 2018, 00:31
The area = π*r^2 = 81π, >> radius = 9units.
Hence the required length of minor arc BC = [(180-110)/360] *[2*π*9] = 7π/2 = 3.5π (Option A)
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In the figure above, AC is the diameter of a circle with center O. If [#permalink]

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12 Apr 2018, 07:20
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Bunuel wrote:

In the figure above, AC is the diameter of a circle with center O. If the area of the circle is 81π, what is the length of minor arc BC?

A. 3.5π
B. 6π
C. 7π
D. 8.5π
E. 9π

[Reveal] Spoiler:
Attachment:
Arc_Length.png

The length of minor arc BC is a fraction of the circumference.

That fraction is determined by the central angle of sector BOC.

That central angle, also called BOC, lies on a straight line with another angle of 110°
Angle BOC = 70°
Straight line = 180°
(180° - 110°) = 70°

Sector BOC = what fraction of the circle:
$$\frac{Part}{Whole}=\frac{SectorAngle}{360°}=\frac{70}{360}=\frac{7}{36}$$ of circle

Sector BOC = $$\frac{7}{36}$$ of the circle

Circumference of circle?

Find the radius from the area, given
$$A = \pi r^2 = 81\pi$$
$$r^2 = 81$$
$$r = 9$$

So circumference = $$2\pi r=18\pi$$

Arc BC is $$\frac{7}{36}$$ of circumference
$$(\frac{7}{36}*18\pi)=\frac{7}{2}\pi=3.5\pi$$

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Re: In the figure above, AC is the diameter of a circle with center O. If [#permalink]

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16 Apr 2018, 16:18
Bunuel wrote:

In the figure above, AC is the diameter of a circle with center O. If the area of the circle is 81π, what is the length of minor arc BC?

A. 3.5π
B. 6π
C. 7π
D. 8.5π
E. 9π

[Reveal] Spoiler:
Attachment:
Arc_Length.png

Since the area is 81π, the radius is 9, and the circumference is 18π.

Since a straight line has 180 degrees, we see that angle BOC measures 70 degrees. We calculate that angle BOC is 70/360 of the entire circle, so the length of minor arc BC is:

70/360 x 18π = 7/36 x 18π = 3.5π

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Re: In the figure above, AC is the diameter of a circle with center O. If   [#permalink] 16 Apr 2018, 16:18
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