QQuestion
ABCD is a cyclic quadrilateral. Bisectors of ∠DAB and ∠BCD intersect the circle at X and Y respectively. If O be the centre of the circle, find ∠XOY.
✓Answer
Given: The bisector of ∠DAB and ∠BCD intersect the circle at the points X and Y.
To Find : ∠XOY
The angles ∠YAB and ∠YCB subtended by the minor arc YB are on the same segment of the circle.
∴ ∠YAB = ∠YCB = 1\over2 ZBCD —- (1) [ ∴ CY is bisector of ∠BCD]
Again, ∠XAY = ∠XAB + ∠YAB
= 1\over2 ∠BAD + 1\over2 ∠BCD [From (1) we get, ∴ AX is bisector of ∠DAB]
= 1\over2 (∠BAD + ∠BCD)
= 1\over2 × 180° [∴ ABCD is a cyclic quadrilateral]
= 90° ∴ ∠XAY is a semicircular angle.
∴ XY is a diameter and ∠XOY = 180°
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