Constructing Images under Enlargement with Positive and Negative Scale Factors

By the end of this lesson, learners should be able to

1. Construct the image of a geometric figure under an enlargement with a positive whole number linear scale factor.
2. Construct the image of an object figure under a positive fractional linear scale factor using ray projections.
3. Construct the inverted image of an object under a negative linear scale factor.

Key vocabulary

Centre of enlargement

A fixed reference point from which ray projections are drawn during geometric enlargement.

The teacher marked the centre of enlargement before drawing rays through each triangle vertex.

Linear scale factor

The constant multiplier ratio comparing corresponding lengths of an image and its original object.

Multiplying distance by a linear scale factor of 3 gives the new vertex position.

Reduction

An enlargement transformation where the linear scale factor is a fraction between zero and one.

Using a fractional scale factor of half results in a reduction of the triangle.

Ray projection

A straight construction line extended from the centre of enlargement through an object vertex.

Each vertex distance was measured accurately along its ray projection line.

Negative scale factor

A scale factor less than zero that projects the image onto the opposite side.

Under a negative scale factor, point $A^{\mathrm{\prime }}$ lies on the opposite side of $O$.

Inverted image

An image turned upside down or reversed across the centre of enlargement due to negative scaling.

The negative scale factor produced an inverted image on the far side of centre $O$.

Lesson preview

Slide 1 · Constructing Images with Positive Integer Scale FactorsOpening · 12 min

To construct the image of a geometric shape under an enlargement with a positive scale factor, straight construction lines originate at a chosen centre of enlargement and pass through every vertex. Measuring and multiplying the distance from this fixed point to each original vertex by the linear scale factor determines the exact location of corresponding image points.

• An enlargement transformation rescales an object relative to a chosen centre of enlargement.
• Construction rays connect the chosen centre of enlargement directly with each vertex of the original figure.
• The distance to an image vertex $O{A}^{\mathrm{\prime }}$ equals the linear scale factor $k$ times $OA$.
• Connecting image vertices produces a similar figure with enlarged side lengths proportional to $k$.

Worked example. Identify the given linear scale factor $k=3$ and the distance from the centre of enlargement $O$ to vertex $A$, $OA=4\text{ cm}$. Apply the enlargement rule $O{A}^{\mathrm{\prime }}=k\times OA$ along the straight ray passing through $O$ and $A$. Calculate $O{A}^{\mathrm{\prime }}=3\times 4\text{ cm}=12\text{ cm}$. Mark point $A^{\mathrm{\prime }}$ on the extended ray such that $O{A}^{\mathrm{\prime }}=12\text{ cm}$, and repeat for vertices $B$ and $C$. Answer: $O{A}^{\mathrm{\prime }}=12\text{ cm}$.

Check question: An object vertex $P$ is $3.5\text{ cm}$ from the centre of enlargement $O$. What is the distance $O{P}^{\mathrm{\prime }}$ after an enlargement with linear scale factor 4?

1. $14\text{ cm}$
2. $0.875\text{ cm}$
3. $7.5\text{ cm}$
4. $12\text{ cm}$

Show answer

$14\text{ cm}$. Multiplying the original distance $OP=3.5\text{ cm}$ by the linear scale factor $k=4$ gives $O{P}^{\mathrm{\prime }}=4\times 3.5\text{ cm}=14\text{ cm}$.

Slide 2 · Constructing Images with Positive Fractional Scale FactorsModel · 14 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 10 lessons in this sub-strand

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All 11 lessons in Similarity and Enlargement

1. 1Determining Centre of Enlargement and Linear Scale FactorDetermine the centre of enlargement and the linear scale factor for similar figures
2. 2Finding the Central Point of Dilation and Scale FactorsDetermine the centre of enlargement and the linear scale factor for similar figures
3. 3Constructing Images under Enlargement with Positive and Negative Scale FactorsConstruct the image of an object under an enlargement given the centre of enlargement and the linear scale factor (L.S.F)
4. 4Constructing Images under Geometric EnlargementConstruct the image of an object under an enlargement given the centre of enlargement and the linear scale factor (L.S.F)
5. 5Calculating Area and Volume Scale Factors for Similar FiguresDetermine the area and volume scale factor of different figures and solids
6. 6Determining Area and Volume Scale FactorsDetermine the area and volume scale factor of different figures and solids
7. 7Connecting Length, Surface Area, and Volume RatiosRelate linear scale factor, area scale factor, and volume scale factor in enlargements
8. 8Connecting Linear, Surface Area, and Volumetric Scale Factors in Similar SolidsRelate linear scale factor, area scale factor, and volume scale factor in enlargements
9. 9Applications of Similarity and Enlargement in Real LifeApply similarity and enlargement to real-life situations
10. 10Applying Similarity and Enlargement in Real-Life ContextsApply similarity and enlargement to real-life situations
11. 11Appreciating Similarity and Enlargement in Real-Life ApplicationsAppreciate the use of similarity and enlargement in real-life situations