# 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

Lines and Angles
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Angles in Polygons
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Polygons
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Types of Angles
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Angles in a Triangle
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Quadrilaterals
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Circle Geometry

Circle Definitions
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Alternate Segment
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Intersecting Chords
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Similarity

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Similar Triangles
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Scale Drawings
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Congruent Triangles
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Pythagoras Theorem

Pythagoras’ Theorem
9 Topics

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Pythagorean Triads
7 Topics

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Trigonometry

Finding Sides
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Finding Angles
10 Topics

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Compass Bearings
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True Bearings
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