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In the Given Figure, Pt Touches the Circle with Centre O at Point R. Diameter Sq is Produced to Meet the Tangent Tr at P. Given ∠Spr = X° and ∠Qrp = Y°; (Ii) Write an Expression Connecting X and Y. - 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 given figure, PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent TR at P.
Given ∠SPR = x° and ∠QRP = y°;
Prove that:
(i) ∠ORS = y°
(ii) Write an expression connecting x and y.

Solution

`∠`QRP = `∠`OSR = y (angles in alternate segment)

But OS = OR (Radii of the same circle)
∴`∠`ORS = `∠`OSR = y
∴ OQ = OR (radii of same circle)
∴ OQR = `∠` ORQ=  90° - y …………(i) (Since OR ⊥ PT )
But in Δ PQR ,
Ext `∠` OQR = x  + y ……(i)
From (i) and (ii)
x + y  = 90°- y
⇒ x  + 2 y = 90° 

  Is there an error in this question or solution?

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Solution In the Given Figure, Pt Touches the Circle with Centre O at Point R. Diameter Sq is Produced to Meet the Tangent Tr at P. Given ∠Spr = X° and ∠Qrp = Y°; (Ii) Write an Expression Connecting X and Y. 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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