Theory question 110
Mensuration of Plane ShapesTwo equal chords AB and BC are each 15 cm long. They meet at a point B on a circle with centre O, such that ∠ABC = 120°.
(a) Illustrate the information in a diagram
(b) Calculate, correct to the nearest whole number the: (i) radius of the circle (ii) perimeter of the major segment cut off by the chord AC. [Take π = 22/7]
Show the model answer
(a) Circle with B on it, chords BA = BC = 15 cm, ∠ABC = 120°, chord AC drawn
(b)(i) AC = 2 × 15 sin 60° ≈ 25.98 cm; sine rule: 2r = AC ÷ sin 120° = 30 ⇒ r = 15 cm (triangles OAB and OBC are equilateral)
(ii) The major arc AC (the arc not containing B) subtends 2 × 120° = 240° at the centre
Major arc = 240/360 × 2 × 22/7 × 15 ≈ 62.86 cm; perimeter = 62.86 + 25.98 ≈ 89 cm
The angle at B on the circumference is half the angle subtended at the centre by the opposite arc.