Circles

This chapter studies the properties of chords and angles in a circle. You will prove that equal chords of a circle subtend equal angles at the centre, and conversely, that the perpendicular from the centre to a chord bisects the chord, and that the line joining the centre to the midpoint of a chord is perpendicular to it. You will also prove that equal chords are equidistant from the centre and that chords equidistant from the centre are equal, and use these results with the Pythagoras theorem to find lengths of chords and distances. You will then study angles subtended by arcs. The angle subtended by an arc at the centre is double the angle it subtends at any point on the remaining part of the circle, angles in the same segment are equal, and the angle in a semicircle is a right angle. Finally, you will learn that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees, and its converse.

Detailed Worksheet: Circles Section A - Definitions (10 marks) 1. Define a chord and a diameter of a circle. Which is the longest chord of a circle? (2 marks) 2. What is a segment of a circle? Distinguish between a major segment and a minor segment. (2 marks) 3. Define an arc. What are a major arc and a minor arc? (2 marks) 4. What is a cyclic quadrilateral? State the property of its opposite angles. (2 marks) 5. What are concentric circles? What are congruent circles? (2 marks) Section B - Calculations and Applications (15 marks) 6. A chord of length 24 cm is drawn in a circle of radius 13 cm. Find the distance of the chord from the centre. (3 marks) 7. The distance of a chord from the centre of a circle of radius 10 cm is 6 cm. Find the length of the chord. (3 marks) 8. Two parallel chords of lengths 6 cm and 8 cm are drawn in a circle of radius 5 cm on opposite sides of the centre. Find the distance between the chords. (3 marks) 9. An arc PQ subtends an angle of 130 degrees at the centre O. Find the angle it subtends at a point R on the remaining part of the circle. (3 marks) 10. ABCD is a cyclic quadrilateral in which angle A = 75 degrees and angle B = 110 degrees. Find angles C and D. (3 marks) Section C - Diagrams (10 marks) 11. Draw a circle with centre O and a chord AB. Draw OM perpendicular to AB. Prove that M is the midpoint of AB, giving reasons at every step. (4 marks) 12. Draw a diagram to illustrate that the angle in a semicircle is a right angle, and use the theorem on the angle at the centre to justify it. (3 marks) 13. Draw a circle with two equal chords AB and CD and show their perpendicular distances from the centre. State the theorem that relates these distances. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove that equal chords of a circle subtend equal angles at the centre. (5 marks) 15. Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. (5 marks) 16. Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees. Hence show that if a side of a cyclic quadrilateral is produced, the exterior angle equals the interior opposite angle. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw neat figures for all proofs and write reasons for every step.