Year 9 Mathematics · AC9M9SP02

Enlargement, Similarity, Ratio and Scale

An enlargement multiplies every length from a centre by the same scale factor. Similarity preserves angle measures and shape while lengths, areas and other measures…

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Learning goalsSay it simply

An enlargement multiplies every length from a centre by the same scale factor. Similarity preserves angle measures and shape while lengths, areas and other measures change predictably.

By the end of this lesson, you should be able to:

  • apply enlargements with positive scale factors
  • identify invariant and changing features
  • connect enlargement to similarity, ratio and scale
  • use dynamic geometry to test and explain transformations
Key conceptTeach from the board

Enlargement is controlled by centre and scale factor

Each image point lies on the ray from the centre through the original point, at a distance multiplied by k.

Similarity preserves shape

Corresponding angles remain equal and corresponding lengths remain in a constant ratio.

Not every measurement scales the same way

Lengths scale by k; areas scale by k².

Coordinates can make enlargement precise

With centre at the origin, (x,y)→(kx,ky). Other centres require measuring from the centre rather than blindly multiplying coordinates.

Worked examplesWe do

Worked examples

AC9M9SP02 - Enlargement, Similarity, Ratio and Scale
Example 1

Origin-centred enlargement k=3 sends (2,−1) to (6,−3).

Example 2

Length scaling A 5 cm side enlarged by k=1.6 becomes 8 cm.

Example 3

Area scaling If k=2, area becomes 4 times as large.

Example 4

Invariant angles A 40° angle remains 40° after enlargement.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9SP02: apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain aspects that remain the same and those that change

Victoria: VC2M9SP02 — Level 9 Space. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Geometrical properties and figures / Ratios and rates. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9SP02VC2M9SP02 — Level 9 SpaceStage 5 Core — Geometrical properties and figures / Ratios and rates
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
What defines an enlargement?
A centre and a scale factor.
What stays the same?
Corresponding angle measures and overall shape.
How does area scale?
By the square of the length scale factor.
Practice and reviewReady for practice
  • Enlargement always means bigger: 0<k<1 produces a reduction.
  • Area scales by k: Area scales by k².
  • Multiply coordinates by k for any centre: That shortcut works only when the centre is the origin.
  • Similar shapes must be the same size: Congruent shapes are same size; similar shapes may be scaled.
  1. [5 marks] Construct an enlargement from a centre and scale factor and state three invariant or changing properties.
  2. [6 marks] Solve a similarity problem involving length and area scale factors.
  3. [7 marks] Compare two enlargements with different centres and use coordinate or dynamic evidence to explain invariants and changes.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

Curriculum alignmentStart here

An enlargement multiplies every length from a centre by the same scale factor. Similarity preserves angle measures and shape while lengths, areas and other measures change predictably.

By the end of this lesson, you should be able to:

  • apply enlargements with positive scale factors
  • identify invariant and changing features
  • connect enlargement to similarity, ratio and scale
  • use dynamic geometry to test and explain transformations
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