Question

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

1. ∠CBT
2. ∠BAT
3. ∠APT

   

Answer 1

 
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°

∴ ∠PTB = 150°

Now in ΔPTB

∠APT + ∠PTB + ∠ABT = 180°

`=>` ∠APT + 150° + 20° = 180°

`=>` ∠APT = 180° – (150° + 20°)

`=>` ∠APT = 180° – 170°

`=>` ∠APT = 10°