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  • Two chords AB and AC of a circle subtends angles equal to 90º and 150º, respectively at the centre. Find angle BAC, if AB and AC lie on the opposite sides of the centre.

Two chords AB and AC of a circle subtend angles equal to 90º and 150º, respectively at the centre. Find  \angleBAC, if AB and AC lie on the opposite sides of the centre.

Answers (1)

Given: Two chords AB and AC of a circle subtend angles equal to 90º and 150º, respectively at the centre.

AB and AC lie on opposite sides of the centre

In \triangleBOA        

OB = OA (Both equal to radius)

\angleOAB = \angleOBA … (i) (angles opposite to equal sides in a triangle are equal)

In \triangleOAB,

\angleOBA + \angleBAO + \angleAOB = 180°  (angle sum property of a triangle)

\Rightarrow \angleOAB + \angleOAB + 90° = 180°  [From equation (1)]

\Rightarrow 2\angleOAB = 180° – 90°

\Rightarrow \angleOAB = \frac{90^{\circ}}{2}= 45°

Now in \triangleAOC, AO = OC  (Both equal to radius)

\angleOCA = \angleOAC … (ii)    (angles opposite to equal sides in a triangle are equal)

Also,

\angleAOC + \angleOAC + \angleOCA = 180°  (angle sum property of a triangle)

150° + 2 \angleOAC = 180°                     

2\angleOAC = 180° – 150°

2\angleOAC = 30°

\angleOAC = 15°

\angleBAC = \angleOAB +\angleOAC = 45° + 15° = 60°

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