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प्रश्न
In the adjoining diagram TA and TB are tangents, O is the centre. If ∠ PAT = 35° and ∠ PBT = 40°.
Calculate:
(i) ∠ AQP, (ii) ∠ BQP
(iii) ∠ AQB, (iv) ∠ APB
(v) ∠ AOB, (vi) ∠ ATB
उत्तर
(i) ∠ AQP = ∠ PAT = 35° ....( Angles are in alternate segment)
(ii) ∠ BQP = ∠ PBT = 40° ....( Angles are in alternate segment)
(iii) ∠ AQB = ∠ AQP + ∠ BQP
∠ AQB = 35° + 40° = 75°
(iv) ∠ APB + ∠ AQB = 180° ....(Opposite ∠s of a cyclic quadrilateral are supplementary)
∴ ∠ APB + 75° = 180°
∴ ∠ APB = 105°
(v) ∠ AOB = 2∠ AQB = 2(75°) = 150° ....(Angle at the centre = 2 Angle at the circumference)
(vi) In quadrilateral AOBT:
∠ ATB = 360° - (∠OAT + ∠OBT + ∠AOB)
∠ ATB = 360° - (90° + 15° + 90°) = 30° ....(∠OAT = ∠OBT = 90° radius, ⊥ tangent).
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