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# In the Following Figure, Pq is the Tangent to the Circle at A, Db is the Diameter and O is the Centre of the Circle. If ∠Adb = 30° and ∠Cbd = 60°, Calculate: (I) ∠Qab, (Ii) ∠Pad, (Iii) ∠Cdb, - 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 is the tangent to the circle at A, DB is the diameter and O is the centre of the circle. If ∠ADB = 30° and ∠CBD = 60°, Calculate:
(i) ∠QAB, (ii) ∠PAD, (iii) ∠CDB,

#### Solution

i) PAQ is a tangent and AB is the chord.
∠QAB = ∠ADB = 30° (angles in the alternate segment)
ii) OA = OD (radii of the same circle)
∴ ∠OAD  =∠ODA = 30°
But, OA ⊥ PQ
∴ ∠PAD = ∠ OAP  - ∠ OAD = 90°  - 30° = 60°
iii) BD is the diameter.
∴ ∠BCD = 90 (angle in a semi – circle)
Now in ΔBCD,
∠ CDB + ∠CBD + ∠BCD  =180°
⇒ ∠ CDB  + 60° + 90°  =180°
⇒ ∠CDB =180° - 150° = 30°

Is there an error in this question or solution?

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Solution In the Following Figure, Pq is the Tangent to the Circle at A, Db is the Diameter and O is the Centre of the Circle. If ∠Adb = 30° and ∠Cbd = 60°, Calculate: (I) ∠Qab, (Ii) ∠Pad, (Iii) ∠Cdb, 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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