(a)(i)(a) Ans: Diameter
Line BOD passes through the center O and connects two points on the circumference, making it a diameter.
(a)(i)(b) Ans: Tangent
Line ABC touches the circle at exactly one point (B) and is perpendicular to the radius at that point.
(a)(ii) Reasons for ∠AOB = 62°:
- The angle between a tangent (AC) and radius (OB) is 90° (∠OBA = 90°)
- In triangle AOB: 180° – 90° – 28° = 62° (sum of angles in triangle)
(a)(iii) Ans: Vertically opposite angles are equal
∠DOE and ∠AOB are vertically opposite angles formed by intersecting lines, hence equal.
(a)(iv)(a) Ans: ∠DEB = 90°
Angle subtended by diameter (DB) at circumference (E) is always 90°.
(a)(iv)(b) Ans: ∠ODE = 59°
In isosceles triangle ODE (OD = OE as radii):
∠ODE = (180° – 62°)/2 = 59°
(a)(iv)(c) Ans: ∠BEF = 149°
∠OEB = ∠ODE = 59° (alternate segment theorem)
∠BEF = 180° – 31° = 149° (angles on straight line AOEF)
(b) Properties of regular polygon:
- All sides are equal in length
- All interior angles are equal
(c) Ans: 144°
Interior angle = (n-2)×180°/n = (10-2)×180°/10 = 144°
