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Consider a circle with area 81π . An arc on this circle is such that i

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Bunuel
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Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 01:44
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If angle ABC is 40 degrees (see figure), and the area of the circle is \(81\pi\), how long is arc AXC? (CB is a diameter of the circle)

A. \(\frac{\pi}{2}\)
B. \(\pi\)
C. \(2\pi\)
D. \(4\pi\)
E. \(8\pi\)


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D
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 03:04
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Concept: Length of an arc =[ø/360]*2*pi*r --------------- (1)
ø - is the central angle say COA, where O is the center of the circle!

Now central ∟COA = 2*∟CAB = 80 degrees ---------------- (2)

Area of circle = π*r*r = 81*π
Therefore r*r = 81
r = 9 -------------- (3)

From 1, 2 & 3
Length of arc AXC = (80/360)*2*π*9 = 4π

Hence D !!
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 03:57
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Bunuel wrote:

If angle ABC is 40 degrees (see figure), and the area of the circle is \(81\pi\), how long is arc AXC? (CB is a diameter of the circle)

A. \(\frac{\pi}{2}\)
B. \(\pi\)
C. \(2\pi\)
D. \(4\pi\)
E. \(8\pi\)


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Property-1: Length of the Arc = (Angle/360)*2*π*r

So we need to calculate the angle made by Arc AXC at the centre of the Circle

Property-2: Angle made at the centre by any arc is always twice the angle made at the circumference by the same arc

i.e. Since Arc AXC makes an angle of 40 degrees at the circumference so
i.e. Arc AXC will make an angle of 2*40 = 80 degrees at the Centre of the circle

i.e. Length of the Arc AXC = (80/360)*2*π*9 [Area = πr^2 = 81π i.e. Radius = 8]

i.e. Length of the Arc AXC = 4π

Answer: Option D
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 04:14
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IMO D is correct
Length of arc AXC= angle subtended at centre/360 * 2 *pi*radius
Angle at centre=2 * angle at circumference=80
Radius
Area=pi* r^2=81pi
Radius=9
Length of arc= 80/360 * 2* pi* 9= 4pi
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 04:32
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Bunuel wrote:

If angle ABC is 40 degrees (see figure), and the area of the circle is \(81\pi\), how long is arc AXC? (CB is a diameter of the circle)

A. \(\frac{\pi}{2}\)
B. \(\pi\)
C. \(2\pi\)
D. \(4\pi\)
E. \(8\pi\)


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GIven area of the circle = 81*\(\pi\)

Thus, \(\pi\)\(r^2\) = 81*\(\pi\)
r=9

NOw, as any angle subtended by an arc on the circumference is half the angle subtended by the same arc at the center of the circle, thus the angle subtended by arc AXC at the center of the circle is 2*40 deg.= 80 deg.

For the complete circumference, 2\(\pi\) or 360 is subtended by 2\(\pi\) r of the circumference. 80 deg will thus subtend = 80/360 * 2\(\pi\) r = 4\(\pi\). thus D is the answer.
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 06:13
So in other words the central angle is twice the size of an arc angle?
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 07:06
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noTh1ng wrote:
So in other words the central angle is twice the size of an arc angle?


Yes, the angle made by the arc is half the angle made by the SAME arc at the center.

In the attached figure, the same arc is minor arc AB.
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   02 Jul 2015, 09:41
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Area =81 pi
R=9
Arch AXC forms the angle at the center =twice the angle formed at circumference= 2*40 =80 degree
Length of Arc AXC=Angle formed at center/360 * 2 pi * R

80/360*2.Pi.9=4.pi

Ans is D
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   03 Jul 2015, 09:10
does this mean ... whenever we are calculating length of arc we need to know angle subtended by it at centre ... not at circumference ..
Right ?
I calculated it as 2*pi
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   03 Jul 2015, 09:23
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adityadon wrote:
does this mean ... whenever we are calculating length of arc we need to know angle subtended by it at centre ... not at circumference ..
Right ?
I calculated it as 2*pi


Yes, the angle always has to be the one that the arc subtends at the center of the circle.

the direct formula for calculating the length of any arc of a circle = l = \(r*\theta\) , where \(\theta\) is the angle measure at the center of the circle in RADIANS and not in degrees and r is the radius of the circle.

\(2\pi\) radians equal 360 degrees.

In the question above, 80 degrees will be equal to \((80/360) * 2\pi\) radians. Radius, r = 9.

Thus the length of the minor arc = \(r*\theta\) = \(9*(80/360) * 2\pi\) = \(4\pi\)
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   06 Jul 2015, 04:34
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Bunuel wrote:

If angle ABC is 40 degrees (see figure), and the area of the circle is \(81\pi\), how long is arc AXC? (CB is a diameter of the circle)

A. \(\frac{\pi}{2}\)
B. \(\pi\)
C. \(2\pi\)
D. \(4\pi\)
E. \(8\pi\)


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MANHATTAN GMAT OFFICIAL SOLUTION:

If the area of the circle is \(81\pi\), then the radius of the circle is 9 (\(A = \pi r^2\)). Therefore, the total circumference of the circle is \(18\pi\) (\(C = 2\pi r\)). Angle ABC, an inscribed angle of 40°, corresponds to a central angle of 80°. Thus, arc AXC is equal to 80/360 = 2/9 of the total circumference:

\(\frac{2}{9}*18\pi = 4\pi\).

Answer: D.
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Re: Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   25 Mar 2016, 23:01
If angle ABC is 40 degrees (see figure), and the area of the circle is 81π, how long is arc AXC? (CB is a diameter of the circle)

pi*r^2 =81 pi
r=9
Angle at the center of the arc can be found as shown in fig...

for 80degree arc
Total perimeter*80/360
2pi*r*80/360
4pi
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Consider a circle with area 81π . An arc on this circle is such that i [#permalink] New post   01 Apr 2016, 20:15
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Chiragjordan wrote:
Consider a circle with area 81π.An arc on this circle is such that it makes a 40 degree inscribed angle with a point P on the circle. What is the length of this arc?
[A] 2π
4π
[C] 8π
[D] 12π
[E] Cannot be determined


Hi,

[b]A GOOD Q.. +1kudos
Difficulty level should 600-700...

1) first, since we are talking of arc length, lets change the Area to Perimeter..
\(A=81π=π*r^2\)
so r=9 and P= 18π

2) Next would be to find the angle that this arc makes at center..
Since the ARC makes 40 degree angle at a point on the circumference, ARC will make 2*40 at the center..

3) Finally lets correlate the angle and perimeter to find length of ARC..
360 degree makes the perimeter or 18π..
so 1 degree will make \(\frac{18π}{360}\), and
2*40 or 80 degree will make\(\frac{18π}{360} * 80 = 4π\)
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Re: Consider a circle with area 81 . An arc on this circle is such that i [#permalink] New post   18 Jun 2023, 07:20
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