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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.
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B
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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: B

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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   06 Sep 2017, 03:47
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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
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post   10 Sep 2019, 09:16
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As a starting point, one might want to look at answer B and consider when would that chord be half of the diameter. We know that a radius would be half the diameter. Given that, we can start looking in that direction. I think that's another start, if someone is stuck, onto the path Greenlight laid out.
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post   17 Sep 2021, 13:11
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answer is B
that is because the length of AB = 2 CD/2sin (Q/2) Q is the angle of AB
as will as AB = CD/2 so Q =60
AS will as the arc CA= BD so their inner angles is
therefore 180 -Q = 2CA angle
this will lead to CA angle = 60 and this can't be as the question stated that Arce AB is smaller than CD not equal
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post   20 Dec 2021, 22:30
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Another way that is more intuitive for me:

2 conditions:
- BD > AB
- BC = AC

Therefore BD = (CD - AB)/2 and since BD > AB, (CD - AB)/2 > AB. Simplifying CD - AB > 2AB -> CD > 3AB.
Sub CD > 3AB into quantity A which is AB/CD -> 1AB/GT3AB -> simplifies to 1/MT3, which is smaller than quantity B.
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post   29 Aug 2022, 22:55
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chord AB=z
chord AC=BD=Y
chord CD=x
let, z be maximum by making, z=y
then,
x=3y

length of chord ab/length of chord cd
= y/x
=y/3y
=1/3
=0.33
at max quantity A is 0.33 which is less than 0.5
so quantity B is larger
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Re: In the semicircle above, the length of arc AC is equal to t [#permalink] New post   13 Feb 2024, 17:02
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we want to AB/CD also AB=CD-2AC
so substituting this we will get CD-2ac/CD
=> cd/cd-2ac/cd
lets think what will be the fraction of 2ac in cd, from the given conditions it is clear it is greater than 2/3
so lets look at possible maximum value and minimum value
=>1-2.1/3 or 1-2.9/3 value is only decrrasing and is lessa than 1 /2 so B is greater
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