Enlargement, Similarity, Ratio and Scale — AC9M9SP02

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

Recall coordinates, ratios, scale factors, similar shapes, angle properties and basic transformations.

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.

• 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. Enlarge triangle coordinates by k=2 about the origin.
2. Reduce a shape using k=0.5.
3. A similar figure has length factor 3. Find the area factor.
4. Use dynamic geometry to verify corresponding angles remain equal.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Map (−3,4) under k=2.5 about the origin.
2. A 12 cm side reduces to 7.2 cm. Find k.
3. If k=0.4, find the area scale factor.
4. Explain similarity versus congruence.
5. Describe geometrically how to enlarge a point about centre C.
6. Explain why angle measures stay invariant under enlargement.

Two students enlarge the same triangle with scale factor 2 but use different centres. Explain which properties of the two images must be the same and which positions can differ.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

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.

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.

• I can use centre and scale factor correctly.
• I can handle reductions as well as enlargements.
• I can distinguish length and area scaling.
• I can explain similarity invariants.
• I can use software to test, not replace, geometric reasoning.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

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.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9SP01 — Why Sine, Cosine and Tangent Are Constant Ratios
• AC9M9M05 — Direct Proportion, Rates, Ratio and Scale Modelling

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Apply the enlargement transformation to shapes and objects using dynamic...

Mapped skill: 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

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.