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

Question

https://gmatclub.com/forum/download/file.php?id=27126 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 Kudos for a correct solution . 2015-07-02_1243.png

NickHalden · Answer

Concept:  Length of an arc = *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 !!

GMATinsight · Answer

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

i.e. Length of the Arc AXC = 4π

Answer: Option D

kanigmat011 · Answer

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

ENGRTOMBA2018 · Answer

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.

noTh1ng · Answer

So in other words the central angle is twice the size of an arc angle?

ENGRTOMBA2018 · Answer

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.

sunita123 · Answer

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

adityadon · Answer

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

ENGRTOMBA2018 · Answer

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*	heta  , where  	heta  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*	heta  =  9*(80/360) * 2\pi  =  4\pi

Bunuel · Answer

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.

Vinayprajapati · Answer

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

chetan2u · Answer

Hi,

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

bumpbot · Answer

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

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