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Intern  Joined: 16 Oct 2010
Posts: 6
In the figure below, AB is the chord of a circle with center  [#permalink]

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1 00:00

Difficulty:   65% (hard)

Question Stats: 53% (02:44) correct 47% (02:34) wrong based on 55 sessions

### HideShow timer Statistics In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

Attachments probpic.JPG [ 6.68 KiB | Viewed 26848 times ]

Intern  Joined: 08 Oct 2010
Posts: 4

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In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

ans : OB=BC
thus, in the triangle BOC, angle BOC = angle BCO =y
again, OB=AO (as both are radii of the same circle)
thus, in triangle OAB, angleOAB= angle OBA =y (because OAB and BOC are equal triangle)
thus, we can make the equation like this,
180=y+(180-2y)+x
thus, y=x
thus, ans is
c
Retired Moderator Joined: 02 Sep 2010
Posts: 755
Location: London

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honeyhani wrote:
In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

OB=BC, So BCO=BOC=x
ABO=2x (exterior angle of triangle BOC, sum of the two opposite interior angles)
OAB=ABO=2x (Isosceles triangle, 2 sides are the radius of the circle)
AOD is an exterior angle of triangle AOC
Hence AOD=OAB+BCO=x+2x=3x=y

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honeyhani wrote:
In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

Sol:
OB = BC (given) => Angle BOC = Angle BCO = y => Angle OBC = 180-2y (in triangle OBC)
Hence Angle OBA = 2y
Since AO = OB (Both radius of the circle)
so Angle OBA = Angle BAO = 2y => Angle AOB = 180 -4y
Now we know all the three angles at point O form by a straight line DC
Hence Angle AOD + Angle AOB + Angle BOC = 180
x + 180 - 4y + y = 180
x - 3y = 0
x = 3y
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Re: In the figure below, AB is the chord of a circle with center  [#permalink]

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honeyhani wrote:
In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

OA:A
Attachment: xy.PNG [ 54.81 KiB | Viewed 4066 times ]

Joining $$OB$$,$$\angle BOC =\angle BCO =y$$ (As $$OB=BC$$)
$$\angle OBA = \angle BOC + \angle BCO$$ (An exterior angle of a triangle is equal to the sum of the opposite interior angles.)
$$\angle OBA = 2y$$
$$\angle OAB = \angle OBA =2y$$ ( As $$OA=OB=r$$)
Considering $$\triangle OAC$$,
$$\angle DOA =\angle OAC + \angle OCA$$ (An exterior angle of a triangle is equal to the sum of the opposite interior angles.)
$$x =y +2y$$
$$x=3y$$ so $$k=3$$
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Joined: 04 Aug 2010
Posts: 436
Schools: Dartmouth College
In the figure below, AB is the chord of a circle with center  [#permalink]

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Quote:
In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3
B. 2
C. 1
D. None of the above

OA and OB are radii, so OA=OB.
Since the prompt indicates that OB=BC, we get:
OA=OB=BC.
The following figure is yielded: Let y=10.
Since the angles inside triangle BCO must sum to 180, and the angles opposite sides OB and BC must be equal, the following figure is yielded: Since angles ABO and OBC must sum to 180, angle ABO=20.
Since the angles inside triangle ABO must sum to 180, and the angles opposite sides OA and OB must be equal, the following figure is yielded: Since angles AOD, AOB and BOC must sum to 180, we get: Since x=30, y=10, and $$k = \frac{x}{y}$$, we get:
$$k = \frac{30}{10} = 3$$

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Available for tutoring in NYC and long-distance. In the figure below, AB is the chord of a circle with center   [#permalink] 20 Aug 2018, 11:45
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