Mathematics

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate :

(i) ∠CBT

(ii) ∠BAT

(iii) ∠APT

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate :  (i) ∠CBT  (ii) ∠BAT (iii) ∠APT. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

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Answer

Join AT and BT.

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate :  (i) ∠CBT  (ii) ∠BAT (iii) ∠APT. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

(i) TC is the diameter of the circle.

Since, angle in a semi-circle is a right angle.

∴ ∠CBT = 90°.

Hence, ∠CBT = 90°.

(ii) In cyclic quadrilateral ATBC,

⇒ ∠CBT + ∠CAT = 180° (∵ Sum of opposite angles of a cyclic quadrilateral = 180°)

⇒ 90° + ∠CAT = 180°

⇒ ∠CAT = 180° - 90°

⇒ ∠CAT = 90°.

In △ABC,

⇒ ∠CBA + ∠CAB + ∠ACB = 180° [By angle sum property of triangle]

⇒ 70° + ∠CAB + 50° = 180°

⇒ ∠CAB + 120° = 180°

⇒ ∠CAB = 180° - 120°

⇒ ∠CAB = 60°.

From figure,

∠BAT = ∠CAT - ∠CAB = 90° - 60° = 30°.

Hence, ∠BAT = 30°.

(iii) From figure,

∠BTX = ∠BAT = 30° [Angle in same segment are equal]

∠PBT = ∠CBT - ∠CBA = 90° - 70° = 20°.

⇒ ∠PTB = 180° - ∠BTX = 180° - 30° = 150°.

In △PBT,

⇒ ∠PBT + ∠PTB + ∠APT = 180° [By angle sum property of triangle]

⇒ 20° + 150° + ∠APT = 180°

⇒ ∠APT + 170° = 180°

⇒ ∠APT = 180° - 170°

⇒ ∠APT = 10°.

Hence, ∠APT = 10°.

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