A, B and C are three points on a circle with centre O. The tangent at A meets BC produced at T. If \(\angle BTA=42^\circ\) and \(\angle CAT=46^\circ\), what is the angle subtended by the chord BC at the centre?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 92°
By the alternate segment theorem, the angle between the tangent AT and the chord AC equals the angle that AC makes in the other segment, \(\angle ABC\). The angle at the centre is then twice the angle at the circumference on the same chord.
\(\angle ABC=\angle CAT=46^\circ\)
In \(\triangle ABT\): \(\angle BAT=180^\circ-42^\circ-46^\circ=92^\circ\)
\(\angle BAC=\angle BAT-\angle CAT=92^\circ-46^\circ=46^\circ\)
\(\angle BOC=2\times\angle BAC=2\times46^\circ=92^\circ\)
