- Mathematics Extension 1
- Year 11
Plane and Circle Geometry
approx. 16 hrs
8 topics
42 concepts
Develop mastery of Plane and Circle Geometry and maximise your HSC results in Mathematics Extension 1.
approx. 16 hrs
8 topics
42 concepts
Develop mastery of Plane and Circle Geometry and maximise your HSC results in Mathematics Extension 1.
This learning program covers the essential components of the Year 11 Mathematics Extension 1 Plane and Circle Geometry topic as outlined the NSW syllabus. We help students gain competence in understanding congruence and similarity and its application to geometrical proofs. This includes providing deductive reasoning and geometrical justification for each step. In addition, we focus on selecting and applying appropriate circle theorems to solve geometrical proofs.
This learning program is made up of the following 8 topics, broken down into 42 concepts.
Application of circle theorems
Tangents from an external point are equal.
The products of the intercepts of intersecting secants are equal
The line joining the centers of two circles passes through their point of contact
The products of the intercepts of two intersecting chords are equal, and conversely
The Tangent-Secant Theorem
Equal chords subtend equal arcs on a circle, and conversely
A line through the centre of a circle perpendicular to a chord bisects the chord, and conversely
Equal arcs subtend equal angles at the centre of a circle, and conversely
Chords equidistant from the centre of a circle are equal, and conversely
Equal chords subtend equal angles at the centre and conversely
Angles at the circumference in the same segment are equal, and conversely
The angle between a tangent and a chord equals the angle at the circumference in the alternate segment
A radius (diameter) of a circle is perpendicular to the tangent at their point of contact
The angle at the circumference in a semi-circle is 90Â°, and conversely
Equal arcs subtend equal angles at the circumference, and conversely
The angle at the centre is twice the angle at the circumference standing on the same arc
Proving Quadrilaterals are Cyclic
The opposite angles of a cyclic quadrilateral are supplementary, and conversely
The exterior angle of a cyclic quadrilateral equals the interior opposite (or remote) angle, and conversely
Proving Concyclicity
Prove and apply tests for quadrilaterals using similarity results
Apply the intercepts and parallel lines properties to find unknown sides
Solve problems involving similarity ratios and areas and volumes
Apply similar triangle results to solve practical problems
Application of congruent triangles to prove properties of special triangles and quadrilaterals
Choosing a test for congruence (SSS test, SAS test, AAS test, RHS test)
Write formal proofs of congruence of triangles, preserving matching order of vertices
Write formal proofs of similarity of triangles, preserving matching order of vertices
Tests for similar triangles
Prove Pythagoras' theorem using similarity
Converse of Pythagoras' theorem
Apply rule that if two lines are parallel to a third line then they are parallel to one another
Use angle relationships to identify parallel lines
Find angles at a point using angle relationships
Use angle relationships of parallel lines to find the size of unknown angles, giving reasons
Apply Exterior Angle Sum of Polygons
Apply the Interior Angle Sum of a Polygon formula
Investigate and apply the exterior angle of a triangle property to find angles in a given diagram
Investigate and use the angle sum of a triangle property to find angles in a given diagram
Apply the triangle inequality rule
Find unknown sides and angles in a given diagram by using the properties of special triangles and quadrilaterals
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