The length of arc AXB is twice

[700+]

The length of arc AXB is twice the length of arc BZC, and the length of arc AYC is three times the length of arc AXB. What is the measure of angle BCA?

1) 20
2) 40
3) 60
4) 80
5) 120

Source: Manhattan CAT II

According to the question,
Length of arc AXB is twice the length of arc BZC => arc AXB = 2 times arc BZC ----> (1)
Length of arc AYC is three times the length of arc AXB => arc AYC = 3 times AXB ---->(2)

From (1), it could be said that 3 * arc AXB = 6 * arc BZC ---->(3)

From (2) & (3), arc AYC : arc AXB : arc BZC = 1 : 3 : 6

Manhattan Explanation Given:
According to the question, the ratio of the lengths of arc AYC to arc AXB to arc BZC is 6: 2: 1.

Could you please tell me what I'm doing wrong over here.

[tim]

According to your calculations, AYC is the smallest arc and BZC is the largest. This goes against the information in the problem. Check your work and let us know if you need any further help..

[nick21]

Hi all
I'm ending up with BZC=40=====> <Z=20
since AXB = 2BZC ===> AXB=80
AYC=6BZC=====> AYC=240
if BCA=BZC+AYC====>BCA=280 ===>> c=140 which is far above any answer choice
where am I wrong please???

[tim]

Nick, i hate to do this, but there are in fact SO many errors in your calculations that i’m going to ask you to take a look at this one again before we help you. The main reason i say this is because most of your mistakes appear to involve a lack of attention to detail - writing incorrect letters that then propogate through your work and make the answers wrong. Please try again and we’ll be glad to help you..

[tanibansal]

2x=40 signifies angle made by the arch at the center of the circle. So angle BCA of the triangle ABC is 20
Answer should be A.

[jnelson0612]

2x=40 signifies angle made by the arch at the center of the circle. So angle BCA of the triangle ABC is 20
Answer should be A.

Are you sure? Unfortunately, that is not the correct answer. Please look again and let us know if you have questions.

[tanibansal]

I am sure of my answer. I got this problem in Manhattan GMAT-3 but got wrong. The angle that is made by the arc is at the center not at the circle and the angle made at the circle by the same arc is half of that made at the centre.

I think the OA should be changed to A.

[tim]

tanibansal, Jamie was trying to say, in as polite a way as possible, that you're wrong and that we can't help you until you show us more of your work so we can identify where you went wrong. your choice at this point is to do what we ask so we can help you, or continue to insist that you're right so we can ignore any further posts from you on the subject..

[tanibansal]

I apologize but i never meant to insult.

I meant, Length of arc is proportional to the angle it subtends at the center. So BXA subtends 80 at the center and hence 40 at the circle.

I got my mistake, a silly one. Once again, I am extremely sorry.

[tim]

no worries at all. sometimes all it takes is to look at the question with an open mind to see where you may have gone wrong. let me know if you have any further questions on this one, and i'll be glad to help..

[jhanasab]

Can you please walk me through how to set up the ratio of 6: 2: 1. I know I am doing it wrong, but I am having a hard time pinpointing what it is. I am getting the ratio 6:3:1.

Thanks for the help.

[jlucero]

Can you please walk me through how to set up the ratio of 6: 2: 1. I know I am doing it wrong, but I am having a hard time pinpointing what it is. I am getting the ratio 6:3:1.

Thanks for the help.

The length of arc AXB is twice the length of arc BZC, and the length of arc AYC is three times the length of arc AXB.

AXB = 2 BZC
AYC = 3 AXB

For simplicity, let's just use the points on the arc:

x = 2z
y = 3x

There's no one perfect way of solving this, but I'd just plug in a number for the smallest arc (z).

z = 1
x = 2
y = 6