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# M13-25

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Math Expert
Joined: 02 Sep 2009
Posts: 48061

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16 Sep 2014, 00:49
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Difficulty:

95% (hard)

Question Stats:

30% (01:22) correct 70% (01:47) wrong based on 94 sessions

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Points $$A$$, $$B$$, $$C$$ and $$D$$ lie on a circle of radius 1. If $$x$$ is the length of arc $$AB$$ and $$y$$ the length of arc $$CD$$ respectively, such that $$x \lt \pi$$ and $$y \lt \pi$$, is $$x \gt y$$?

(1) $$\angle ADB$$ is acute

(2) $$\angle ADB \gt \angle CAD$$

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16 Sep 2014, 00:49
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Official Solution:

Statement (1) by itself is insufficient.

Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.

Statements (1) and (2) combined are sufficient. If $$\angle$$ ADB is acute and greater than $$\angle$$ CAD, then arc $$AB$$ is greater than arc $$CD$$.

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Joined: 14 Mar 2015
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13 Sep 2015, 00:24
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient.

Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.

Statements (1) and (2) combined are sufficient. If $$\angle$$ ADB is acute and greater than $$\angle$$ CAD, then arc $$AB$$ is greater than arc $$CD$$.

Choice B is very tempting !!
I got it wrong
But, thanks for the explanation.
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27 Jan 2016, 00:03
Statements (1) and (2) combined are sufficient. If ∠ ADB is acute and greater than ∠ CAD, then arc AB is greater than arc CD.

Can you elaborate on this?
Also where do we use the fact that both arcs are < pi?
I feel like this has some very complicated geometry involved. :S
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Joined: 17 Sep 2014
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Concentration: Operations, Strategy
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06 Aug 2016, 06:01
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient.

Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.

Statements (1) and (2) combined are sufficient. If $$\angle$$ ADB is acute and greater than $$\angle$$ CAD, then arc $$AB$$ is greater than arc $$CD$$.

Hi Bunuel,

Can you please explain how combining both statements will help in taking a decision that AB>CD ?
Math Expert
Joined: 02 Sep 2009
Posts: 48061

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06 Aug 2016, 11:04
avdgmat4777 wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient.

Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.

Statements (1) and (2) combined are sufficient. If $$\angle$$ ADB is acute and greater than $$\angle$$ CAD, then arc $$AB$$ is greater than arc $$CD$$.

Hi Bunuel,

Can you please explain how combining both statements will help in taking a decision that AB>CD ?

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Joined: 19 Apr 2016
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17 Aug 2016, 01:50
Hi Bunuel,

Since x and y both are arcs and less than pie, and as radius is 1, hence both the arcs are less than a semi circle. Now combining this with ∠ADB>∠CAD, it means both are acute angles, because these could have been 90 degree if they would have been making a semicircle arc or larger if the arc would have been larger, but neither is the case. Hence the answer should be B.

Am I thinking right or missing something?
Math Expert
Joined: 02 Sep 2009
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17 Aug 2016, 02:05
bjain01 wrote:
Hi Bunuel,

Since x and y both are arcs and less than pie, and as radius is 1, hence both the arcs are less than a semi circle. Now combining this with ∠ADB>∠CAD, it means both are acute angles, because these could have been 90 degree if they would have been making a semicircle arc or larger if the arc would have been larger, but neither is the case. Hence the answer should be B.

Am I thinking right or missing something?

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Joined: 26 Dec 2017
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16 Jul 2018, 04:05
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient.

Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.

Statements (1) and (2) combined are sufficient. If $$\angle$$ ADB is acute and greater than $$\angle$$ CAD, then arc $$AB$$ is greater than arc $$CD$$.

Hi BUnuel,
One doubt here how can we measure ADB whether it is obtuse or acute is there any property in circle.
Especially in above two figures how the angle difference is spotted?
Pls explain
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Re: M13-25 &nbs [#permalink] 16 Jul 2018, 04:05
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# M13-25

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