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Math Expert V
Joined: 02 Sep 2009
Posts: 58315
If O is the center of the circle shown above and BC=CD, what is the va  [#permalink]

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11 00:00

Difficulty:   25% (medium)

Question Stats: 75% (01:29) correct 25% (02:19) wrong based on 138 sessions

HideShow timer Statistics 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°

Attachment: T6085.png [ 10.7 KiB | Viewed 1949 times ]

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Current Student 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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Is the answer 20? Sorry Bunuel, I cannot see the answer since I am using the mobile app

Sent from my iPhone using GMAT Club Forum mobile app
Math Expert V
Joined: 02 Sep 2009
Posts: 58315
Re: If O is the center of the circle shown above and BC=CD, what is the va  [#permalink]

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ProfX wrote:
Is the answer 20? Sorry Bunuel, I cannot see the answer since I am using the mobile app

Sent from my iPhone using GMAT Club Forum mobile app

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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I have opted for 20 deg.
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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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Opting for 20 degrees.
Intern  Joined: 23 Jun 2016
Posts: 41
Re: If O is the center of the circle shown above and BC=CD, what is the va  [#permalink]

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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
Current Student 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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How do we know that ACD is 90 degrees?
Intern  Joined: 23 Jun 2016
Posts: 41
If O is the center of the circle shown above and BC=CD, what is the va  [#permalink]

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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.
Retired Moderator G
Joined: 26 Nov 2012
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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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1
Bunuel wrote: 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°

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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ACD=90

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
hence x=20

Posted from my mobile device
Intern  B
Joined: 05 Sep 2015
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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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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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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°
Attachments 1.png [ 22.33 KiB | Viewed 1522 times ]

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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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1
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 $$X = 20$$
Attachments 14747092178945.png [ 55.57 KiB | Viewed 1467 times ]

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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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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]

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_________________ Re: If O is the center of the circle shown above and BC=CD, what is the va   [#permalink] 02 Jan 2019, 02:39
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