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Math Expert V
Joined: 02 Sep 2009
Posts: 58435

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Difficulty:   95% (hard)

Question Stats: 32% (01:51) correct 68% (02:38) wrong based on 76 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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Math Expert V
Joined: 02 Sep 2009
Posts: 58435

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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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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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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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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 V
Joined: 02 Sep 2009
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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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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 V
Joined: 02 Sep 2009
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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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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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82vkgmat wrote:
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

I think the fact that both arcs are < pi is mentioned cuz we are suppose to consider only the minor arcs (less than half the perimeter) of the 2 points and not the major arcs.

The reason it is vital is because the result of comparing minor arcs of 2 sets of points would be opposite of comparing the result of 2 major arcs of the same set of points.
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In gmat club test, this question is under below 500 level, are gmat 500 level questions are really tough like this?
Math Expert V
Joined: 02 Sep 2009
Posts: 58435

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mangamma wrote:
In gmat club test, this question is under below 500 level, are gmat 500 level questions are really tough like this?

This is 750+ question. Where is it listed as sub-500?
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