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Points A,B,C lie on the circle. Point O is the center of the circle an

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Points A,B,C lie on the circle. Point O is the center of the circle an  [#permalink]

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New post 08 Jan 2015, 00:32
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A
B
C
D
E

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Points A,B,C lie on the circle. Point O is the center of the circle and lies on the line segment AB. If angle COB is 100 degrees then angle CAO equals to

A. 90 degrees
B. 80 degrees
C. 60 degrees
D. 50 degrees
E. 40 degrees

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Re: Points A,B,C lie on the circle. Point O is the center of the circle an  [#permalink]

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New post 08 Jan 2015, 01:28
ANS: D

OC & OB are radii, hence triangle OCB is an isosceles triangle.
Angle OCB= Angle OBC
Angle OCB+Angle OBC+ Angle COB = 180
Angle COB = 100
Angle OCB = Angle OBC = (180 - 100) /2 = 40deg

Now, with reference to a theorem which states that any triangle inscribed in a circle with diameter as one of the sides, the opposite angle to diameter is 90 deg.

Therefore angle ACB = 90 deg
ACB = 90, Angle CBA = 40
Hence Angle CAO = 180 - (90+40) = 50 Deg.
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Re: Points A,B,C lie on the circle. Point O is the center of the circle an  [#permalink]

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New post 08 Jan 2015, 14:48
Hi All,

You don't actually need to figure out the value of big Angle C to answer this question.

We're told that angle COB = 100 degrees.

Since lines add up to 180 degrees, this means that angle COA = 80 degrees

Triangle AOC is ISOSCELES, since OA and OC are both radii of the circle. This means that the two missing angles in that triangle are the SAME.

Triangle AOC = X + X + 80 degrees = 180 degrees

2X + 80 = 180
2X = 100
X = 50 degrees

Angle CAO = 50 degrees

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Re: Points A,B,C lie on the circle. Point O is the center of the circle an  [#permalink]

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New post 18 Jul 2019, 12:58
Hey isn't that a perpendicular bisector? Hence, mustn't angle AOC be 90 degrees?? Getting confused here.
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Re: Points A,B,C lie on the circle. Point O is the center of the circle an  [#permalink]

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New post 18 Jul 2019, 14:44
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BjornMattsson wrote:
Hey isn't that a perpendicular bisector? Hence, mustn't angle AOC be 90 degrees?? Getting confused here.


the diagram says COB is 100 so the other AOC cannot be 90 rather 80 because a straight line is 180 degree .

the below links are great . give it a try

https://gmatclub.com/forum/math-circles-87957.html

https://gmatclub.com/forum/math-triangles-87197.html
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Re: Points A,B,C lie on the circle. Point O is the center of the circle an   [#permalink] 18 Jul 2019, 14:44
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