It is currently 20 Feb 2018, 13:40

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# The length of minor arc AB is twice the length of minor arc

Author Message
TAGS:

### Hide Tags

Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 527
Location: United Kingdom
GMAT 1: 730 Q49 V45
GPA: 2.9
WE: Information Technology (Consulting)
The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

11 Feb 2012, 13:55
14
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

64% (01:18) correct 36% (01:54) wrong based on 254 sessions

### HideShow timer Statistics

The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120

[Reveal] Spoiler:
Attachment:

Arcs.PNG [ 7.21 KiB | Viewed 15216 times ]
[Reveal] Spoiler: OA

_________________

Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Math Expert
Joined: 02 Sep 2009
Posts: 43831
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

11 Feb 2012, 14:18
Expert's post
5
This post was
BOOKMARKED
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.

[Reveal] Spoiler:
Attachment:

Arcs.PNG [ 7.21 KiB | Viewed 17225 times ]

_________________
Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 527
Location: United Kingdom
GMAT 1: 730 Q49 V45
GPA: 2.9
WE: Information Technology (Consulting)
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

11 Feb 2012, 14:26
How did you get x+2x+6x=360 Bunuel?
_________________

Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Math Expert
Joined: 02 Sep 2009
Posts: 43831
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

11 Feb 2012, 14:27
enigma123 wrote:
How did you get x+2x+6x=360 Bunuel?

Circumference 360 degrees.
_________________
Manager
Status: Employed
Joined: 17 Nov 2011
Posts: 96
Location: Pakistan
GMAT 1: 720 Q49 V40
GPA: 3.2
WE: Business Development (Internet and New Media)
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

15 Feb 2012, 12:08
1
KUDOS
1
This post was
BOOKMARKED
Let BC $$=x$$
So AB $$=2x$$
So AC $$=6x$$

Now angle in question is opposite AB... and we know that Angles are in proportion to the length of the sides they are in front of..

We need to find $$2x$$.....so ratio of this side(AB) $$=\frac{{2x}}{{9x}}=\frac{2}{9}$$

Sum of angles of triangle $$=180$$

So we can use ratios to setup an equation as follows:

$$\frac{{2x}}{{9x}}*180=40$$

Hence Angle is $$40$$
_________________

"Nowadays, people know the price of everything, and the value of nothing." Oscar Wilde

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7942
Location: Pune, India
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

16 Feb 2012, 03:26
4
KUDOS
Expert's post
1
This post was
BOOKMARKED
enigma123 wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

Any idea how to solve this guys?

Given the ratio of minor arcs, their central angles and inscribed angles are in the same ratio.
arc BC: arc AB : arc AC = 1:2:6
So inscribed angles angle A : angle C : angle B = 1:2:6
Since these three angles form a triangle, their sum is 180 which means angle C = 40 degrees
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 21 Oct 2011 Posts: 4 Re: The length of minor arc AB is twice the length of minor arc [#permalink] ### Show Tags 16 Feb 2012, 04:01 lenght of the arc = (x/360)*2pir Given AB=2BC; AC=3AB. If x is the angle of arc BC then AB=2x; AC=6x; AB+BC+AC=360 --> x+2x+6x=360 --> 9x = 360 --> x = 40 Is this approach right? can someone help me please? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7942 Location: Pune, India Re: The length of minor arc AB is twice the length of minor arc [#permalink] ### Show Tags 16 Feb 2012, 04:41 pavan1mpv wrote: lenght of the arc = (x/360)*2pir Given AB=2BC; AC=3AB. If x is the angle of arc BC then AB=2x; AC=6x; AB+BC+AC=360 --> x+2x+6x=360 --> 9x = 360 --> x = 40 Is this approach right? can someone help me please? When you say, "If x is the angle of arc BC", you mean the central angle, right? The sum of the central angles will be 360 so you get x = 40. But remember, this is the central angle subtended by arc BC. The central angle subtended by arc AB = 2x = 80. But since we are interested in the inscribed angle (i.e. the angle on the circumference of the circle), we get angle BCA (inscribed angle of arc AB) = 80/2 = 40 degrees (since inscribed angle is half of central angle) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Intern
Joined: 16 Dec 2011
Posts: 46
GMAT Date: 04-23-2012
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

22 Feb 2012, 07:15
as all arcs are given so we can add all arcs to find radius of the circle bcoz sum of all arcs is equal to circumference of the circle,

assume BC = X,
AB= 2X AND AC= 6X
implies 9X = 2pir
implies, r= 9X/2pi
now angle subtended by arc AB at the centre of circle is twice of the angle BCA.

arc AB = 2X = angle at centre of circle ( Q)/360 *2pir

implies 2X= Q/360*
Intern
Joined: 16 Dec 2011
Posts: 46
GMAT Date: 04-23-2012
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

22 Feb 2012, 07:18
sorry submitted uncompleted ,

2X= Q/360*2pi*9X/2PI
Gives us angle Q= 80
and angle BCA = 40 ( HALF OF THE ANGLE subtended at centre of circle)

Non-Human User
Joined: 09 Sep 2013
Posts: 13815
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

15 Jul 2014, 13:07
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.
_________________
Intern
Joined: 10 May 2015
Posts: 29
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

03 Jul 2015, 00:36
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.
Math Expert
Joined: 02 Sep 2009
Posts: 43831
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

03 Jul 2015, 01:17
davesinger786 wrote:
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.

Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414
_________________
Intern
Joined: 10 May 2015
Posts: 29
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

03 Jul 2015, 01:42
Bunuel wrote:
davesinger786 wrote:
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.

Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414

Thanks for that,Bunuel. So if I go by the logic that the diagram isn't accurate, arc AC subtends 240 at the center right? but in this question ,it's mentioned as the "minor arc AC" w.r.t to the calculations but going by the diagram in your article it looks like 240 is actually made by the major arc? Please help me get an idea of what are the angles subtended by the arcs ..getting confused here.Thanks in advance
Senior Manager
Joined: 15 Sep 2011
Posts: 358
Location: United States
WE: Corporate Finance (Manufacturing)
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

03 Jul 2015, 10:39
davesinger786 wrote:
Bunuel wrote:
davesinger786 wrote:

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.

Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414

Thanks for that,Bunuel. So if I go by the logic that the diagram isn't accurate, arc AC subtends 240 at the center right? but in this question ,it's mentioned as the "minor arc AC" w.r.t to the calculations but going by the diagram in your article it looks like 240 is actually made by the major arc? Please help me get an idea of what are the angles subtended by the arcs ..getting confused here.Thanks in advance

That's true. If AC is larger than the two other archs, AC cannot be described as a minor arch. The archs are 2x their inscribed angles, and therefore, divide the degree of the arch by 2 to arrive at your answer.

Thanks,
Board of Directors
Joined: 17 Jul 2014
Posts: 2729
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE: General Management (Transportation)
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

14 Dec 2016, 14:59
enigma123 wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

Any idea how to solve this guys?

what a nice question!
i got to the answer the same way
9x=360
x=40
2x=80
and as explained, the answer is B.
Intern
Joined: 19 May 2017
Posts: 1
Re: The length of minor arc AB is twice the length of minor arc [#permalink]

### Show Tags

26 Aug 2017, 09:46
[quote="davesinger786"][quote="Bunuel"]The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.

would be thankful if someone could explain the above quoted discrepancy in the problem
Re: The length of minor arc AB is twice the length of minor arc   [#permalink] 26 Aug 2017, 09:46
Display posts from previous: Sort by