QQuestion
In the given figure, O is the centre of the circle. PQ is a tangent to the circle at T. Chord AB produced meets the tangent at P. AB = 9 cm, BP = 16 cm, ∠PTB = 50° and ∠OBA = 45°. Find:
(a) the length of PT
(b) ∠BAT
(c) ∠BOT
(d) ∠ABT
✓Answer
(a) PAB is a secant and PT is a tangent from P.
By the tangent-secant theorem:
PT2 = PA × PB
PA = PB + BA = 16 + 9 = 25 cm
PT2 = 25 × 16
PT2 = 400
PT = √400
Therefore, PT = 20 cm.
(b) By the alternate segment theorem, the angle between tangent PT and chord TB equals the angle subtended by chord TB in the alternate segment.
Therefore, ∠BAT = ∠PTB = 50°.
(c) The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference.
∠BOT = 2 × ∠BAT
∠BOT = 2 × 50°
Therefore, ∠BOT = 100°.
(d) OB = OT, since both are radii.
Let ∠OBT = ∠OTB = x.
In △BOT:
x + x + 100° = 180°
2x = 80°
x = 40°
Thus, ∠OBT = 40°.
∠ABT = ∠ABO + ∠OBT
∠ABT = 45° + 40°
Therefore, ∠ABT = 85°.
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