uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures
generalises mathematical ideas and techniques to analyse and solve problems efficiently
uses deductive reasoning in presenting arguments and formal proofs
applies deductive reasoning to prove circle theorems and to solve related problems
- identify and name parts of a circle (centre, radius, diameter, circumference, sector, arc, chord, secant, tangents, segment, semicircle)
- use terminology associated with angles in circles, eg subtend, standing on the same arc, angle at the centre, angle at the circumference, angle in a segment
- identify the arc on which an angle at the centre or circumference stands
- demonstrate 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
- prove the following chord properties of circles:
- chords of equal length in a circle subtend equal angles at the centre and are equidistant from the centre
- the perpendicular from the centre of a circle to a chord bisects the chord; conversely, the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord
- the perpendicular bisector of a chord of a circle passes through the centre
- given any three non-collinear points, the point of intersection of the perpendicular bisectors of any two sides of the triangle formed by the three points is the centre of the circle through all three points
- when two circles intersect, the line joining their centres bisects their common chord at right angles
- use dynamic geometry software to investigate chord properties of circles (Problem Solving, Reasoning)
- prove the following angle properties of circles:
- the angle at the centre of a circle is twice the angle at the circumference standing on the same arc
- the angle in a semicircle is a right angle
- angles at the circumference, standing on the same arc, are equal
- the opposite angles of cyclic quadrilaterals are supplementary
- an exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle
- use dynamic geometry software to investigate angle properties of circles (Problem Solving, Reasoning)
- apply circle theorems to prove that the angle in a semicircle is a right angle (Problem Solving, Reasoning)
- apply chord and angle properties of circles to find unknown angles and lengths in diagrams
- Prove and apply tangent and secant properties of circles
- prove the following tangent and secant properties of circles:
- the two tangents drawn to a circle from an external point are equal in length
- the angle between a tangent and a chord drawn to the point of contact is equal to the angle in the alternate segment
- when two circles touch, their centres and the point of contact are collinear
- the products of the intercepts of two intersecting chords of a circle are equal
- the products of the intercepts of two intersecting secants to a circle from an external point are equal
- the square of a tangent to a circle from an external point equals the product of the intercepts of any secants from the point
- use dynamic geometry software to investigate tangent and secant properties of circles (Problem Solving, Reasoning)
- apply tangent and secant properties of circles to find unknown angles and lengths in diagrams
As well as solving arithmetic and algebraic problems in circle geometry, students should be able to reason deductively within more theoretical arguments. To aid reasoning, they should be given diagrams labelled with the important information. Students should also be able to produce a clear diagram from a set of instructions.
Attention should be given to the logical sequence of theorems and to the types of arguments used. The memorisation of proofs is not intended. Ideally, every theorem presented should be preceded by a straight-edge-and-compasses construction to confirm the theorem, before it is proven in an appropriate manner, by way of an exercise or an investigation.
The tangent-and-radius theorem is difficult to justify in Stage 5.3 and is probably better taken as an assumption, as indicated above.
This substrand may be extended to examining the converse of some of the theorems related to cyclic quadrilaterals, leading to a series of conditions for points to be concyclic. However, students may find these results difficult to prove and apply.
The angle in a semicircle theorem is also called Thales' theorem because it was traditionally ascribed to the philosopher Thales of Miletus (c624–c546 BC) by the ancient Greeks, who reported that it was the first theorem ever proven in mathematics.
Purpose/Relevance of Substrand
The study and application of angle relationships and the properties of geometrical figures, undertaken in Stage 4 and Stage 5, is taken further to the context of the circle for those students who study the substrand Circle Geometry. These students add a further dimension to their knowledge, skills and understanding in geometry by learning to analyse circle geometry problems and by developing a broader set of geometric and deductive reasoning skills, as well as additional problem-solving skills. They develop an understanding that the geometry of the circle is very important in the work of architects, engineers, designers, builders, physicists, etc, as well as in everyday situations through its occurrence in nature, sports, buildings, etc.
A considerable amount of specific terminology is introduced in Stage 5.3 Circle Geometry. Teachers will need to model and explain the correct use of terms such as 'subtend', 'point of contact', 'collinear', 'standing on the same arc' and 'angle in the alternate segment'.
Students should write geometrical reasons without the use of abbreviations to assist them in learning new terminology and in understanding and retaining geometrical concepts. If students abbreviate geometrical reasons that they use in circle geometry, they must take care not to abbreviate the reasons to such an extent that the meaning is lost.