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Maths Quest 10 for New South Wales 5.3 Pathway Work Programs Chapter 15 Circle geometry Strand: Space and geometry Substrands and outcomes: Properties of geometrical figures Properties of geometrical figures Properties of geometrical figures Deductive geometry Circle geometry Section SGS4.3 SGS4.4 SGS5.2.2 SGS5.3.1 SGS5.3.4 Classifies, constructs and determines the properties of triangles and quadrilaterals Identifies congruent and similar two-dimensional figures stating the relevant conditions Develops and applies results for proving that triangles are congruent or similar Constructs arguments to prove geometrical results Applies deductive reasoning to prove circle theorems and to solve problems GC tips, Investigations, History of mathematics, Maths Quest challenge, 10 Quick Questions, Code puzzles, Career profiles SkillSHEETS, WorkSHEETS, Interactive games, Test yourself, Topic tests (CD ROM) Technology applications (CD ROM) Learning outcomes Are you ready? (page 542) Introduction (page 543) Angles in a circle (page 545) WE 1, 2, 3 Ex 15A Angles in a circle (page 549) SkillSHEETs (page 542) 15.1: Using tests to prove congruent triangles 15.2: Corresponding sides and angles of congruent triangles 15.3: Using tests to prove similar triangles 15.4: Angles in a triangle 15.5: More angle relations 15.7: Calculating arc length Investigation: Angles in a circle (page 545) Investigation: Constructing a tangent (page 547) Maths Quest challenge Q1-3 (page 551) SkillSHEET 15.1: Using tests to prove congruent triangles (page 543) SkillSHEET 15.2: Corresponding sides and angles of congruent triangles (page 543) SkillSHEET 15.3: Using tests to prove similar triangles (page 543) SkillSHEET 15.4: Angles in a triangle (pages 549) Cabri geometry: Angles at the centre and the circumference (page 545) Cabri geometry: Angles at the centre and the circumference – proof (page 545) Cabri geometry: Angles subtended by the same arc (page 546) Cabri geometry: Tangents SGS5.2.2 applying the congruency tests to justify that two triangles are congruent SGS4.4 matching sides and angles of two congruent polygons SGS5.2.2 determining whether triangles are similar SGS4.3 applying geometric facts, properties and relationships to solve numerical problems such as finding unknown sides and angles in diagrams SGS5.3.1 constructing and writing geometrical arguments to prove a general geometrical result, giving reasons at each step of the argument proving statements about geometrical figures (Reasoning, Communicating, Applying strategies) SkillSHEET 15.5: More angle relations: (page 549) WorkSHEET 15.1 (page 551) (page 547) SGS5.3.4 identifying and naming parts of a circle (centre, radius, diameter, circumference, sector, arc, chord, secant, tangent, segment, semicircle) using terminology associated with angles in circles such as subtend, standing on the same arc, angle at the centre, angle at the circumference, angle in a segment identifying the arc on which an angle at the centre or circumference stands demonstrating that at any point on a circle, there is a unique tangent to the circle, and that this tangent is perpendicular to the radius at the point of contact applying circle theorems to prove that the angle in a semicircle is a right angle Intersecting chords, secants and tangents (page 552) WE 4, 5, 6, 7 Ex 15B Intersecting chords, secants and tangents (page 557) Investigation: Intersecting chords (page 552) Investigation: Intersecting secants (page 553) History of mathematics: (page 559) Code puzzle (page 560) Investigation: Quadrilaterals in circles (page 561) Cyclic quadrilaterals (page 561) WE 8, 9 Ex 15C Cyclic quadrilaterals 10 Quick Questions 1 (page 564) SkillSHEET 15.6: Angle sum of a quadrilateral (page 563) Game time 001 (page 564) WorkSHEET 15.2 (Applying strategies, Reasoning) applying circle theorems to find unknown angles and sides in diagrams (Applying strategies, Reasoning) constructing tangents to a circle (Applying strategies) Cabri geometry: SGS5.3.4 Intersecting chords using the tangent to a (page 552) circle result as an Cabri geometry: assumption when Intersecting secants proving theorems (page 553) involving tangents Cabri geometry: proving and applying Intersecting tangents the theorems on: (page 554) - chord properties Cabri geometry: Chords - angle properties (page 555) - tangents and secants applying circle theorems to find unknown sides in diagrams (Applying strategies, Reasoning) Cabri geometry: Cyclic SGS5.3.4 quadrilaterals I proving and applying (page 561) the theorems on: Cabri geometry: Cyclic - chord properties quadrilaterals II – proof - angle properties (page 563) Tangents, secants and chords (page 565) WE 10, 11 Ex 15D Tangents, secants and chords (page 567) (page 564) Maths Quest challenge Q1-2 (page 570) Investigation: Circle constructions (page 570) (page 561) Cabri geometry: Exterior angles of cyclic quadrilaterals (page 562) Cabri geometry: Alternate segment theorem (page 565) - tangents and secants applying circle theorems to find unknown angles in diagrams (Applying strategies, Reasoning) using circle and other theorems to prove geometrical results and in problem solving (Applying strategies, Reasoning) SGS5.3.4 using the tangent to a circle result as an assumption when proving theorems involving tangents proving and applying the theorems on: - chord properties - angle properties - tangents and secants applying circle theorems to find unknown angles and sides in diagrams (Applying strategies, Reasoning) using circle and other theorems to prove geometrical results and Great circles (page 571) Investigation: Greenwich WE 12, 13, 14 Mean Time (page 572) Ex 15E Great circles (page Investigation: SOS! 575) (page 576) 10 Quick Questions 2 (page 577) Maths Quest challenge Q1-2 (page 577) Investigation: Eratosthenes calculates the diameter of the Earth! (page 578) Summary (page 579) Chapter review (page 580) SkillSHEET 15.7: Calculating arc length (page 575) WorkSHEET 15.2 (page 576) Game time 002 (page 576) ‘Test yourself’ multiple choice questions Topic tests (2) Mathcad: Arc length (page 575) in problem solving (Applying strategies, Reasoning) finding the centre of a circle by construction (Applying strategies) constructing tangents to a circle (Applying strategies) using circle and other theorems to prove geometrical results and in problem solving (Applying strategies, Reasoning)