Question

An angle subtended at the centre of a circle by an arc, is divided into two parts by the radius through the mid-point of the arc. The relation between the two angles

• A. cannot be determined
• B. is unequal
• C. is equal
• D. Is both A & B.

Answer 1

Answer: (C)

is equal

Given- O is the centre of a circle of which AQB is an arc. Q is the midpoint of arc AQB. OQ is the radius of the same circle through the midpoint Q of the arc AQB.
To find out- the relation between $\angle AOQ\&\angle BOQ=?$
Solution- We join AQ & BQ. Since Q is the midpoint of the arc AQB, we get $arcArQ=arcQsB$
$⟹$ the $chordAQ=chordBQ$(because chords, contained by equal arcs, are equal).
So between $\Delta AOQ$ & $\Delta BOQ$ we have $AO=QO=BO$(radii of the same circle), AQ=BQ. \therefore By SSS test, we have $\Delta AOQ\cong \Delta BOQ⟹\angle AOQ=\angle BOQ.$
So OQ bisects $\angle AOB.$
Ans- Option C.