VCE Specialist Mathematics Units 1 and 2 – Geometry, Measurement and Trigonometry

4.1 Geometry
4.2 Circle Geometry
4.3 Similarity
4.4 Logic
4.5 Pythagoras Theorem
4.6 Trigonometry
4.7 Non-Right Angled Triangles
4.8 Vectors

Geometry in the plane and proof

This topic includes:

  • geometric objects and relations: point, line, parallel, perpendicular, plane, angle, polygons, circles and semicircles, arcs, chords, segments, sectors, secants, tangents, similarity and congruence
  • straight edge and compass and dynamic geometry construction of these objects and illustration of these relations, including exact angles multiples of \( 30^{\circ} \) and \( 45^{\circ} \)
  • principles of proof including propositions and quantifiers, examples and counter-examples, direct proof, proof by contradiction, and proof using contrapositive; and the role of diagrams in a geometric proof
  • proofs of Pythagoras’ theorem, properties of quadrilaterals, interior angles and angle sums of polygons
  • congruence of triangles and the sine and cosine rules including applications
  • proof of circle theorems such as:
    • the angle at the centre subtended by an arc (chord) of a circle is twice the angle at the circumference subtended by the same arc (chord), including the case of the semi-circle and right angle
    • angles at the circumference of a circle subtended by the same arc (chord) are equal
    • the opposite angles of a cyclic quadrilateral are supplementary
    • chords of equal length subtend equal angles at the centre and conversely, chords subtending equal angles at the centre of a circle have the same length
    • the alternate segment theorem
    • results about intersecting chords where the chords intersect inside or outside the circle (as secants) including the limiting case, where one of the lines is a tangent
    • converses of some of the above results.

Vectors in the plane

This topic includes:

  • representation of plane vectors as directed lines segments, examples involving position, displacement and velocity
  • magnitude and direction of a plane vector, and unit vectors
  • geometric representation of addition, subtraction (triangle and/or parallelogram rules) scalar multiple and linear combination of plane vectors
  • representation of a plane vector as an ordered pair \( (a, b) \) and as a column matrix \( \displaystyle \pmatrix{a \\ b} \)
  • representation of a vector \( (a, b) \) in the form \( \displaystyle a \vec{i} + b \vec{j} = a \pmatrix{1 \\ 0} + b \pmatrix{0 \\ 1} = \pmatrix{a \\ b} \) where \( i \) and \( j \) are the standard orthogonal unit vectors, and direction cosines
  • simple vector algebra (addition, subtraction, multiplication by a scalar, linear combination) using these forms
  • a scalar product of two plane vectors, perpendicular and parallel vectors, projection of one vector onto another, and angle between two vectors
  • application of vectors to geometric proofs, orienteering, navigation, and statics.

source – VCE Mathematics Study Design

VCE Specialist Mathematics Units 1 and 2 Courses

VCE Specialist Mathematics Units 1 and 2 Syllabus

Course Content

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Geometry

Lesson Content
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Circle Geometry
Lesson Content
0% Complete 0/1 Steps
Similarity
Pythagoras Theorem
Trigonometry
Non-Right Angled Triangles
Vectors
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Course Includes

  • 49 Lessons
  • 281 Topics
  • 42 Quizzes