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] |
\[\Rightarrow \]\[2\angle OBC=180{}^\circ -100{}^\circ \] |
\[\Rightarrow \]\[2\angle OBC=80{}^\circ \] |
\[\Rightarrow \]\[\angle OBC=40{}^\circ \] |
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