8.1 Complete the following theorem: The angle subtended by the diameter at the circumference of the circle is .. - NSC Technical Mathematics - Question 8 - 2019 - Paper 2

8.2.1 (a) ∠CBO = 90° - 30° = 60°

Explanation: This follows from the property that the angle subtended by the radius at the circumference equals the angle in the alternate segment.

8.2.1 (b) ∠D₂ = 30°

Explanation: As AB is tangent at B, ∠D₂ is equal to ∠CBO due to the tangent-secant theorem.

8.2.1 (c) ∠O₁ = 60°

Explanation: Since the angle at the center is twice that at the circumference, ∠OBC = 60°.

8.2.1 (d) ∠O₂ = ∠D₂ = 30°

Explanation: By the cyclic nature of quadrilaterals, and since CO || FE, alternate angles are equal.

8.2.2 Since ∠D = 60° and ∠O = 30°, triangle BOC is isosceles with sides FB = FE. Also, DE = CB.

By the property of triangles, the sides opposite equal angles are equal, hence FC = FD.