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The length of arc AXB is twice the length of arc BZC, and

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Joined: 23 Dec 2013
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Re: The length of arc AXB is twice the length of arc BZC, and [#permalink]

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New post 21 May 2017, 16:07
MisterEko wrote:
Attachment:
Arcs.gif

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?

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

[Reveal] Spoiler:
My reasoning was:

AXB=2BCZ
AYC=3AXB
AYC=6BCZ

2BCZ+BCZ+3BCZ=360 Degrees

BCZ=40 degrees

AXB=80 degrees

AYC=240 degrees

Since AXB=80 and is opposite of angle BCA will be half of that (40)
At the same time, since BCZ is 40 degrees, BAC will be 20, and finally, since AYC is 240, ABC will be 120. I am not sure if this rule exists, it just made sense to me, and my answer of 40 degrees was correct. Am i right here?


Here the quickest method is to recognize that the arcs are in proportion to each other. The ratio is 1:2:6:9, with 1 as the smallest arc, BZC, 2 as the second largest arc, AXB, and thus by definition 6 as the largest arc. The angles corresponding to these arcs will be in the same proportion, so 9x = 180 and x = 20, the hidden multiplier. The angle C on the triangle corresponds to the 2 value in the ratio, so the product is 2(20) = 40.

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The length of arc AXB is twice the length of arc BZC, and [#permalink]

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New post 04 Sep 2017, 19:19
Can someone explain something to me here?

When I first looked at this, I noted the proportions: x, 2x, 6x. When you sum it up, you get 9x. However, I set 9x=360. I understand some posts are setting this equation equal to 180 because angles in a triangle sum up to 180 and the question is asking for the measure of an angle in a triangle...but since the triangle is inscribed in the circle, i was confused and set it equal to 360 b/c thats the sum of angles in a circle.

allow me to expand: when i set 9x=360, i get x=40. since angle BCA represents arc AXB, i know the value here is 2x. 2(40) = 80 degrees. HOWEVER, inscribed angles are HALF of the central angle. so what i did was halve 80 to get 40. so...same (correct) answer, different way.

can anyone explain if my way is correct/incorrect? and shed some light into the question i asked above? thank you!

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Re: The length of arc AXB is twice the length of arc BZC, and [#permalink]

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New post 04 Sep 2017, 21:41
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Expert's post
LakerFan24 wrote:
Can someone explain something to me here?

When I first looked at this, I noted the proportions: x, 2x, 6x. When you sum it up, you get 9x. However, I set 9x=360. I understand some posts are setting this equation equal to 180 because angles in a triangle sum up to 180 and the question is asking for the measure of an angle in a triangle...but since the triangle is inscribed in the circle, i was confused and set it equal to 360 b/c thats the sum of angles in a circle.

allow me to expand: when i set 9x=360, i get x=40. since angle BCA represents arc AXB, i know the value here is 2x. 2(40) = 80 degrees. HOWEVER, inscribed angles are HALF of the central angle. so what i did was halve 80 to get 40. so...same (correct) answer, different way.

can anyone explain if my way is correct/incorrect? and shed some light into the question i asked above? thank you!



Sum of the angles of a triangle is ALWAYS 180.
So, angle BAC, angle ABC and angle BCA add up to 180 degrees.

The angles around the centre of a circle will add up to a complete 360 degrees. So in this fig: https://gmatclub.com/forum/the-length-o ... l#p1235120
Angles AOB, BOC and COA add up to 360 degrees.
If you assume that x is angle subtended at the centre by arc BZC, you do get x + 2x + 6x = 360 and subsequently the correct answer.
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The length of arc AXB is twice the length of arc BZC, and [#permalink]

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New post 20 Nov 2017, 11:35
The length of arc AXB is twice the length of arc BZC => angle C = 2 (angle A) (1)
the length of arc AYC is three times the length of arc AXB => angle B = 3 (angle C) (2)
In triangle ABC, angle A + angle B + angle C = 180 degree (3)
From (1) (2) (3) => angle A + 2 (angle A) + 6 (angle A) = 9 angle A = 180 degree
=> angle A = 180/9 = 20 degree => BCA= 20*2 = 40 degree => Answer is B

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The length of arc AXB is twice the length of arc BZC, and   [#permalink] 20 Nov 2017, 11:35

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