In the figure, O is the centre of a circle. If Angle DOB = 90° and AngleDOC = 60°, find the value of Angle BAC
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Class 9
>>Maths
>>Circles
>>Angles in the Same Segment
>>In Fig 10.9 AOB = 90^∘ and ABC = 30^∘
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In Fig 10.9 ∠AOB=90
∘
and∠ABC=30
∘
then∠CAO is equal to
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Easy
Solution
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Correct option is D)
In △AOB
OA=OB
∴∠OAB+∠OBA+∠AOB=180
0
=>2∠OAB+90
0
=180
0
=>2∠OAB=180
0
−90
0
=90
0
=>∠OAB=
2
90
=45
0
We know that the angle subtended at the centre of the circle by an arch is twice the angle subtended at the circumference by the same arc.
∴∠ACB=
2
1
∠AOB
=
2
90
=45
0
Again in △ABC
∠ACB+∠CBA+∠BAC=180
0
=>45
0
+30
0
+∠BAC=180
0
=>∠BAC=180
0
−75
0
=105
0
∴∠CAO=∠BAC−∠OAB
=>∠CAO=105
0
−45
0
=60
0