# 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

### Geometry

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Circle Geometry
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Similarity
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Pythagoras Theorem
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Trigonometry
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Non-Right Angled Triangles
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Vectors
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• 49 Lessons
• 281 Topics
• 42 Quizzes