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If O is the center of the circle shown above and BC=CD, what is the va

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If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 05:09
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If O is the center of the circle shown above and BC=CD, what is the value of x?

A. 15°
B. 20°
C. 25°
D. 30°
E. 40°

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Attachment:
T6085.png
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[Reveal] Spoiler: OA

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 05:20
Is the answer 20? Sorry Bunuel, I cannot see the answer since I am using the mobile app


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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 05:25
ProfX wrote:
Is the answer 20? Sorry Bunuel, I cannot see the answer since I am using the mobile app


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The OA will be automatically revealed on Thursday 29th of September 2016 05:09:24 AM Pacific Time Zone
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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 06:05
I have opted for 20 deg.

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New post 22 Sep 2016, 11:27
Opting for 20 degrees.

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 13:35
I got the answer 20 but had a question for clarification. We know that ACD is 90 but we cannot assume ABC is also 90 since AC and AD are different lengths. Based on my calculations I got CAB = 20, ABC = 110 and BCA = 50. Wanted to know if this was the correct process.

Thanks

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New post 22 Sep 2016, 14:03
How do we know that ACD is 90 degrees?

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New post 22 Sep 2016, 16:09
joannaecohen wrote:
How do we know that ACD is 90 degrees?


A triangle inscribed in a circle who hypotenuse is the diameter of the circle forms a 90 degree angle at the circumference.

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 22:05
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Bunuel wrote:
Image
If O is the center of the circle shown above and BC=CD, what is the value of x?

A. 15°
B. 20°
C. 25°
D. 30°
E. 40°

[Reveal] Spoiler:
Attachment:
T6085.png


This is a tricky question. We have hidden arc and angle information and we can find the angle information via arc.

( NOTE : Arc angle is double the center angle )

ACD is a right angled triangle. we have C = 90 , D (given) = 70 and thus A is 20.

When we have angle A 20 degrees and the arc under CD is 40.

We are given that BC = CD...i.e. arc under BC is also 40 degrees and finally x is 20.

IMO option B.

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 22 Sep 2016, 22:30
ACD=90
Hence CAD=20

also ABC=110 as opp angles of a cyclic quad are supplementary.

join BD
ABD=90
DBC=ABC-ABD=110-90=20

In triangle BCD, CBD=CDB=20

hence BDA=50

now in triangle ABD
ABD=90, BDA=50, BAD=x+20
hence x=20

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 23 Sep 2016, 01:28
In ΔACD, ∠ACD is an inscribed angle intercepting the diameter of the circle and is therefore 90°. ∠CAD=180-(70+90)=20°. Hence, minor arc CD is 40°. Minor arcs BC and CD lie on identical chords and are therefore equal. Thus, minor arc BC is 40°, and x=20°

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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 23 Sep 2016, 04:12
Harish Yannam wrote:
In ΔACD, ∠ACD is an inscribed angle intercepting the diameter of the circle and is therefore 90°. ∠CAD=180-(70+90)=20°. Hence, minor arc CD is 40°. Minor arcs BC and CD lie on identical chords and are therefore equal. Thus, minor arc BC is 40°, and x=20°





this file will explain visually
i hope it helps
Answer is B 20°
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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 24 Sep 2016, 02:54
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From the attached diagram,

Inscribed angle is half of central angle.
Hence \(BAC = 1/2*BOC\)
Likewise \(CAD = 1/2*COD\)

Also since \(BC = CD\), arc BC = arc CD => \(BOC = COD\)
therefore \(BAC = CAD\) that is \(X = Y\)

Now \(OC = OD\) hence \(OCD = ODC\) => \(COD = 40\)

thus \(X = 20\)
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Re: If O is the center of the circle shown above and BC=CD, what is the va [#permalink]

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New post 18 Sep 2017, 03:15
azamaka wrote:
Harish Yannam wrote:
In ΔACD, ∠ACD is an inscribed angle intercepting the diameter of the circle and is therefore 90°. ∠CAD=180-(70+90)=20°. Hence, minor arc CD is 40°. Minor arcs BC and CD lie on identical chords and are therefore equal. Thus, minor arc BC is 40°, and x=20°





this file will explain visually
i hope it helps
Answer is B 20°


Hi,
seems to be a good solution, but I have a question-How can we say that AB=BD in order to get both the green angles as 45 deg?
Thanx

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Re: If O is the center of the circle shown above and BC=CD, what is the va   [#permalink] 18 Sep 2017, 03:15
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