Theory question 113
Circle Theorems(a) In the diagram ABT is a circle centre O. PQ is a tangent to the circle at T and ABC is a straight line. TC bisects ∠BTQ. ∠BAT = 44° and ∠PTA = 60°. Find ∠ACT.
(b) The circumference of the base of a cyclindrical tank is 11 m. The height of the tank is 3 m more than 6 times the base radius. Calculate the: (i) radius; (ii) height; (iii) volume of the tank. [Take π = 22/7]
Show the model answer
Model answer
(a) ∠BTQ = ∠BAT = 44° (alternate segment) ⇒ ∠BTC = 22°
∠ABT = ∠PTA = 60° (alternate segment); ∠ABT is an exterior angle of triangle BTC ⇒ ∠ACT = 60° − 22° = 38°
(b)(i) 2 × 22/7 × r = 11 ⇒ r = 1.75 m
(ii) h = 6 × 1.75 + 3 = 13.5 m
(iii) V = 22/7 × 1.75² × 13.5 ≈ 129.9 m³
Explanation
Use the alternate segment theorem twice.