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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

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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.
i) In quadrilateral ORPQ,
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°

  Is there an error in this question or solution?

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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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