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In the figure shown. point O is the center of the semicircle

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Source: — Data Sufficiency |
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IMO C

by agoyal2 » Sun Nov 30, 2008 12:16 am
IMO C:

Isosceles triangles with two sides = Radius

Angle BOA = Angle BAO
Angle BCO = Angle CBO

From 1: Angle COD = 60.
From 2: Angle BCO = 40 => Angle COB = 100

Angle BAO = Angle BOA = 180-60-100=20

Whats the OA?
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by mals24 » Sun Nov 30, 2008 4:55 am
Ok there are 2 concepts of a triangle that are applied to this question.
1. exterior angle theorem: An exterior angle of a triangle is equal to the sum of the opposite interior angles.
2. opposite sides of equal angles are also equal.

Given in the question:

OC=OB=AB. BAO = ?

St 1: COD = 60

BAO+BCO = 60 (Exterior angle theorem)

BCO = OBC

BAO+OBC = 60

BAO+BOA = OBC (exterior angle theorem)
BAO = BOA

2BAO = OBC

3BAO = 60
BAO = 20....SUFF

ST 2 BCO = 40
BCO = OBC
2BAO = OBC
BAO = 20....SUFF

Answer is D.
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