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In the semicircle above, the length of arc AC is equal to t

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In the semicircle above, the length of arc AC is equal to t [#permalink] New post 19 Jun 2017, 11:52
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In the semicircle above, the length of arc AC is equal to the length of arc BD, and the length of arc AB is less than the length of arc BD.


Quantity A
Quantity B
\(\frac{\text{the length of chord AB}}{\text{the length of chord CD}}\)
\(\frac{1}{2}\)


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post 26 Jul 2017, 15:52
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Any explanation? Thanks in advance!
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post 03 Sep 2017, 08:22
Could someone post an explanation to this question? Would be very appreciated!
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post 05 Sep 2017, 16:43
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Carcass wrote:


Image

In the semicircle above, the length of arc AC is equal to the length of arc BD, and the length of arc AB is less than the length of arc BD.
Quantity A
Quantity B
\(\frac{\text{the length of chord AB}}{\text{the length of chord CD}}\)
\(\frac{1}{2}\)


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.


Okay, so here's what we have so far...
Image
As you can see, I've added the entire circle AND I've added the circle's center

Now the question tells us that the length of arc AB is less than the length of arc BD.
However, at this point, I want to investigate what would happen if it were the case that the length of arc AB is equal to the length of arc BD
We'd get something like this...
Image
Notice that all 3 arcs (AC, AB, and BC) all have the SAME length.

This means that each CENTRAL ANGLE that "holds" these 3 equal arcs must also be equal...
Image

Since all three angles are on the same line (the diameter to be exact), they must add to 180°, which means each angle must be 60°
Image

Since OA and OB are the radii of the circle, we can conclude that ∠OAB and ∠OBA must both be equal, which means they both equal 60°
Image

So, ∆OAB is an EQUILATERAL TRIANGLE, which means all 3 sides have equal length.
In fact all 3 sides are equal to the radius of the circle.
Image

Since OC and OD are also radii, we can see that we have a BUNCH of line segments that are all the same length.
Image

At this point, we can see that: (length of chord AB)/(length of chord CD) = 1/2 [since AB = the length of 1 radius, and CD = the length of 2 radii]

So, if it were the case that the length of arc AB is equal to the length of arc BD, then Quantities A and B would be EQUAL.

However, the original question tells us that the length of arc AB is less than the length of arc BD.
From this, we can conclude that chord AB is LESS THAN the radius of the circle.

This means (length of chord AB)/(length of chord CD) < 1/2

Answer:
[Reveal] Spoiler:
B


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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post 06 Sep 2017, 03:47
Expert's post
Awesome.

Thank you so much.

Regards
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post 13 Mar 2019, 08:37
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Can we explain the problem this way:

As the length of arc AC=BD, So, Chord AB is parallel to CD. Arc drawn on the chord CD is proportionate to the arc drawn on the chord AB. Since, arc AB<BD or AC, hence AB/CD= arc Y (say)/arc AC+BD+Y. so, the ration falls far short of 1/2. Hence, B
Re: In the semicircle above, the length of arc AC is equal to t   [#permalink] 13 Mar 2019, 08:37
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In the semicircle above, the length of arc AC is equal to t

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