The circumcentre of a \[\Delta ABC\] is O. If \[\angle BAC=85{}^\circ \]and \[\angle BCA=75{}^\circ ,\]then the value of\[\angle OAC\]is [SSC (10+2) 2012] |
A) \[40{}^\circ \]
B) \[60{}^\circ \]
C) \[70{}^\circ \]
D) \[90{}^\circ \]
Correct Answer: C
Solution :
\[\because \]\[\angle BAC=85{}^\circ \] |
\[\therefore \]\[\angle BOC=2\times 85{}^\circ =170{}^\circ \] |
[since, angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at any point on the remaining part of circle] |
In \[\Delta BOC,\]\[OB=OC\] [radii of circle] |
So, \[\angle OBC=\angle OCB\] |
Now, \[\angle BOC+\angle OBC+\angle OCB=180{}^\circ \] |
\[\Rightarrow \] \[170{}^\circ +\angle OBC+\angle OBC=180{}^\circ \] |
\[\Rightarrow \] \[2\angle OBC=180{}^\circ -170{}^\circ \] |
\[\Rightarrow \] \[\angle OBC=\frac{10{}^\circ }{2}=5{}^\circ \] |
Now, \[\angle OCB+\angle OCA=75{}^\circ \] |
\[\angle OCA=75{}^\circ -5{}^\circ =70{}^\circ \] |
In \[\Delta AOC,\] \[OC=OA\] [radii of circle] |
so that \[\angle OCA=\angle OAC=70{}^\circ \] |
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