GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Oct 2019, 11:56

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58340
Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 01:44
1
5
00:00

Difficulty:

45% (medium)

Question Stats:

64% (01:46) correct 36% (01:41) wrong based on 75 sessions

### HideShow timer Statistics

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.

Attachment:

2015-07-02_1243.png [ 7.27 KiB | Viewed 5201 times ]

_________________
Manager
Status: Perspiring
Joined: 15 Feb 2012
Posts: 82
Concentration: Marketing, Strategy
GPA: 3.6
WE: Engineering (Computer Software)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 03:04
1
1
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 !!
CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2974
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 03:57
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. $$8\pi$$

Kudos for a correct solution.

Attachment:
2015-07-02_1243.png

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

_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Manager
Joined: 27 Jul 2014
Posts: 235
Schools: ISB '15
GMAT 1: 660 Q49 V30
GPA: 3.76
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 04:14
1
IMO D is correct
Length of arc AXC= angle subtended at centre/360 * 2 *pi*radius
Angle at centre=2 * angle at circumference=80
Area=pi* r^2=81pi
Length of arc= 80/360 * 2* pi* 9= 4pi
CEO
Joined: 20 Mar 2014
Posts: 2603
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 04:32
1
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. $$8\pi$$

Kudos for a correct solution.

Attachment:
2015-07-02_1243.png

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.
Manager
Joined: 07 Apr 2015
Posts: 154
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 06:13
So in other words the central angle is twice the size of an arc angle?
CEO
Joined: 20 Mar 2014
Posts: 2603
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 07:06
2
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.
Attachments

Central Angle Theorem.jpg [ 15.13 KiB | Viewed 4126 times ]

Manager
Joined: 13 Oct 2013
Posts: 134
Concentration: Strategy, Entrepreneurship
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

02 Jul 2015, 09:41
1
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
_________________
---------------------------------------------------------------------------------------------
Kindly press +1 Kudos if my post helped you in any way
Manager
Joined: 18 Mar 2014
Posts: 226
Location: India
Concentration: Operations, Strategy
GMAT 1: 670 Q48 V35
GPA: 3.19
WE: Information Technology (Computer Software)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

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
_________________
Press +1 Kudos if you find this Post helpful
CEO
Joined: 20 Mar 2014
Posts: 2603
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

03 Jul 2015, 09:23
2
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 = $$r*\theta$$ = $$9*(80/360) * 2\pi$$ = $$4\pi$$
Math Expert
Joined: 02 Sep 2009
Posts: 58340
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

06 Jul 2015, 04:34
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. $$8\pi$$

Kudos for a correct solution.

Attachment:
2015-07-02_1243.png

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:

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

_________________
Manager
Joined: 24 May 2013
Posts: 77
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

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
Attachments

Circle.png [ 27.99 KiB | Viewed 3368 times ]

Math Expert
Joined: 02 Aug 2009
Posts: 7957
Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

01 Apr 2016, 20:15
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

[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$$\frac{18π}{360} * 80 = 4π$$
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13172
Re: Consider a circle with area 81π . An arc on this circle is such that i  [#permalink]

### Show Tags

09 Mar 2019, 14:55
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: Consider a circle with area 81π . An arc on this circle is such that i   [#permalink] 09 Mar 2019, 14:55
Display posts from previous: Sort by