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# In the Following Figure, Pq and Pr Are Tangents to the Circle, with Centre O. If ∠Qpr = 60°, Calculate: (I) ∠Qor (Ii) ∠Oqr (Iii) ∠Qsr - ICSE Class 10 - Mathematics

ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

In the following figure, PQ and PR are tangents to the circle, with centre O. If ∠QPR = 60°, calculate:
(i) ∠QOR  (ii) ∠OQR (iii) ∠QSR

#### Solution

Join QR.
OQ  ⊥ OP,OR ⊥  RP
∴ ∠OQP = 90°  ,∠ORP = 90° ,∠QPR = 60°
∠QOR 360° -( 90° + 90° + 60°)
∠QOR = 360° - 240°
∠QOR = 120°

ii) In ∠QOR ,
OQ = QR (Radii of the same circle)
∴ OQR = ∠QRO            ………….(i)
But, ∠OQR +∠QRO+ ∠QOR =180°
∠OQR + ∠ QRO + 120° =  180°
∠ OQR +  ∠QRO 60°
From (i)

2 ∠OQR = 60°
∠ OQR = 30°
iii) Now arc RQ subtends ∠QOR at the centre and ∠QSR at the remaining part of the circle.

∴ ∠ QSR = 1/2 ∠QOR

⇒ ∠QSR = 1/2xx120°

⇒ ∠QSR =60°

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#### Video TutorialsVIEW ALL [1]

Solution In the Following Figure, Pq and Pr Are Tangents to the Circle, with Centre O. If ∠Qpr = 60°, Calculate: (I) ∠Qor (Ii) ∠Oqr (Iii) ∠Qsr Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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