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Angles subtended by opposite sides of anyquadrilateral at the centre of incircle are(a) Complementary Jy Supplementary(d) ObtuseAc) 90° |
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Answer» Let ABCD be a quadrilateral circumscribing a circle with centre O.Now join AO, BO, CO, DO.From the figure, ∠DAO = ∠BAO [Since, AB and AD are tangents] Let ∠DAO = ∠BAO = 1 Also ∠ABO = ∠CBO [Since, BA and BC are tangents] Let ∠ABO = ∠CBO = 2 Similarly we take the same way for vertices C and DSum of the angles at the centre is 360° Recall that sum of the angles in quad. ABCD = 360° ⇒2(1 + 2 + 3 + 4) = 360° ⇒1 + 2 + 3 + 4 = 180° In ΔAOB, ∠BOA= 180 – (a + b)In ΔCOD, ∠COD = 180 – (c + d)Angle BOA + angle COD = 360 – (a + b + c + d)= 360° – 180°= 180°Hence AB and CD subtend supplementary angles at OThus, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. |
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