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Question
PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate:
- ∠CBT
- ∠BAT
- ∠APT
Solution
Join AT and BT.
i. TC is the diameter of the circle
∴ ∠CBT = 90° ...(Angle in a semi-circle)
ii. ∠CBA = 70°
∴ ∠ABT = ∠CBT – ∠CBA
= 90° – 70°
= 20°
Now, ∠ACT = ∠ABT = 20° ...(Angle in the same segment of the circle)
∴ ∠TCB = ∠ACB – ∠ACT
= 50° – 20°
= 30°
But, ∠TCB = ∠TAB ...(Angles in the same segment of the circle)
∴ ∠TAB or ∠BAT = 30°
iii. ∠BTX = ∠TCB = 30° ...(Angles in the same segment)
∴ ∠PTB = 180° – 30° = 150°
Now in ΔPTB
∠APT + ∠PTB + ∠ABT = 180°
`=>` ∠APT + 150° + 20° = 180°
`=>` ∠APT = 180° – (150° + 20°)
`=>` ∠APT = 180° – 170° = 10°
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