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Question

Consider the figure where AOB=90 and ABC=30. Then, CAO = ___.

A
45
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B
50
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C
75
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D
60
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Solution

The correct option is D 60
Given that AOB=90 and ABC=30.
We know that the angle subtended by an arc of a circle at the centre is twice the angle subtended by it at any remaining part of the circle.
AOB=2 ACB
i.e., 90=2ACB
ACB=45
Also, AO=OB.
ABO=BAO [angles opposite to equal sides are equal] .....(i)

In ΔOAB,
OAB+ABO+BOA=180. [angle sum property of a triangle]
OAB+OAB+90=180 [from Eq. (i)]
2OAB=18090
OAB=902=45

Using angle sum property in ΔACB, we have
ACB+CBA+CAB=180.
45+30+CAB=180
CAB=18075=105
But, CAO=CABOAB
=10545=60.
i.e., CAO=60

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