question_answer

In \[\Delta ABC,\]\[O\] is its circumcentre and \[\angle BAC=50{}^\circ .\]The measure of\[\angle OBC\]is                    [SSC (CPO) 2013]

A) \[60{}^\circ \]                          

B) \[30{}^\circ \]

C) \[40{}^\circ \]  

D) \[50{}^\circ \]

Correct Answer: C

Solution :

In \[\Delta ABC,\]

O is the circumcentre of \[\Delta ABC.\]

\[\therefore \]      \[\angle BOC=2\angle BAC\]

[since, angle subtended at the centre is twice the angle subtended at the point A]

\[\therefore \]\[\angle BOC=2\times 50{}^\circ =100{}^\circ \]

Now,     \[\angle OBC=\angle OCB\]

[\[\because \]angle along same side OB = OC are radius]

Now, in \[\Delta BOC\]

\[\Rightarrow \]\[\angle OBC+\angle OCB+\angle BOC=180{}^\circ \]

[angle sum property]

\[\Rightarrow \]\[2\angle OBC=180{}^\circ -100{}^\circ \]

\[\Rightarrow \]\[2\angle OBC=80{}^\circ \]

\[\Rightarrow \]\[\angle OBC=40{}^\circ \]