Enlargements Revision Notes for Leaving Cert Mathematics

Enlargements

An enlargement is a transformation that changes the size of a shape while keeping its proportions exactly the same. The shape becomes bigger or smaller, but its angles remain unchanged and its sides stay in the same ratio to each other.

infoNote

Think of enlarging a photograph - the image becomes bigger, but everything in it maintains the same proportional relationships. This is exactly what happens in mathematical enlargements.

Key terminology

Understanding enlargements requires knowing these essential terms:

Object: The original shape before transformation
Image: The new shape after the enlargement
Centre of enlargement: The fixed point from which the enlargement takes place
Scale factor: The number that tells us how many times bigger (or smaller) the image is compared to the object
Rays or guidelines: The straight lines drawn from the centre of enlargement through corresponding points on the object and image

Scale factor

The scale factor is the most important measurement in enlargements. It tells us the relationship between the size of the image and the size of the object.

chatImportant

Scale factor = length of image side ÷ length of corresponding object side

$\text{Scale factor} = \frac{\text{length of image side}}{\text{length of corresponding object side}}$

For example, if a side of the object measures 3 cm and the corresponding side of the image measures 9 cm, then: Scale factor = $9 \div 3 = 3$

This means the image is 3 times larger than the object.

When we know the scale factor (let's call it $k$ ), we can find any length in the image:

$\text{Length of image side} = k \times \text{length of corresponding object side}$

Drawing enlargements

To construct an enlargement, you need two pieces of information:

The centre of enlargement
The scale factor

lightbulbExample

Step-by-Step Construction Method:

Draw a straight line from the centre of enlargement to each vertex of the object
Measure the distance from the centre to each vertex
Multiply this distance by the scale factor
Mark the new vertex at this calculated distance along the same line
Join up all the new vertices to complete the image

When a vertex is the centre of enlargement

infoNote

Sometimes one of the vertices of the object acts as the centre of enlargement. When this happens, that particular vertex doesn't move - it stays in exactly the same position in both the object and the image.

For all other vertices, you still follow the same construction method, measuring distances from the fixed vertex and multiplying by the scale factor.

Enlargements with scale factor less than 1

When the scale factor is less than 1 (but still positive), the "enlargement" actually produces a smaller figure. This is sometimes called a reduction.

chatImportant

Scale Factor Rules:

If $k > 1$ , the figure gets bigger (true enlargement)
If $k \< 1$ , the figure gets smaller (reduction)
If $k = 1$ , the figure stays the same size

For example, if the scale factor is $\frac{1}{2}$ , then each side of the image will be half the length of the corresponding side in the object.

Finding the centre of enlargement

When you're given both the object and its image, you can find the centre of enlargement using this method:

lightbulbExample

Finding the Centre of Enlargement:

Choose any two pairs of corresponding vertices
Draw a straight line through each pair
Extend these lines until they meet
The point where they meet is the centre of enlargement

This method works because all the rays from the centre of enlargement pass through corresponding points on the object and image.

Enlargement and area

There's a special relationship between enlargements and area that's crucial for exam success:

chatImportant

Area Scaling Rule:

When a figure is enlarged by scale factor $k$ , the area of the image = $k^2 \times$ area of the object

$\text{Area of image} = k^2 \times \text{Area of object}$

Notice that the area scales by $k^2$ , not just $k$ . This is because area involves two dimensions (length and width), so both get multiplied by the scale factor.

lightbulbExample

Worked Example: Area Relationship

If a triangle has an area of 12 cm² and is enlarged by scale factor 2, what's the area of the image?

Solution: Area of image = $k^2 \times$ area of object Area of image = $2^2 \times 12 = 4 \times 12 = 48$ cm²

Worked example: Finding scale factor and lengths

lightbulbExample

Worked Example: Complete Enlargement Analysis

Problem: Triangle ABC is enlarged to triangle A'B'C' with centre of enlargement O. Given |AC| = 6, |CC'| = 9, and |B'C'| = 12.5, find:

The scale factor
The length |BC|
The ratio |AB| : |AB'|

Solution:

(i) Finding the scale factor: First, find |AC'| = |AC| + |CC'| = 6 + 9 = 15

Scale factor = |AC'| ÷ |AC| = $\frac{15}{6} = 2.5$

(ii) Finding |BC|: Since scale factor = 2.5, we know |B'C'| = 2.5 × |BC| 12.5 = 2.5 × |BC| |BC| = $\frac{12.5}{2.5} = 5$

(iii) Finding the ratio |AB| : |AB'|: Since |AB'| = 2.5 × |AB|, the ratio is: |AB| : |AB'| = |AB| : 2.5|AB| = 1 : 2.5 = 2 : 5

Worked example: Area problem

lightbulbExample

Worked Example: Finding Scale Factor from Areas

Problem: Figure P'Q'R'S' is an enlargement of figure PQRS. If the area of PQRS is 12 cm² and the area of P'Q'R'S' is 48 cm², find the scale factor.

Solution: Let $k$ be the scale factor. Area of image = $k^2 \times$ area of object $48 = k^2 \times 12$ $k^2 = \frac{48}{12} = 4$ $k = 2$

Therefore, the scale factor is 2.

Inverse enlargements

infoNote

Sometimes you need to go from the enlarged image back to the original object. This is called an inverse enlargement.

If the original enlargement had scale factor $k$ , then the inverse enlargement has scale factor $\frac{1}{k}$ .

For example, if shape A was enlarged by scale factor $1\frac{1}{2}$ to give shape B, then to go from B back to A, you'd use scale factor $1 \div 1\frac{1}{2} = \frac{2}{3}$ .

bookmarkSummary

Key Points to Remember:

Scale factor = length of image side ÷ length of object side
Area scales by k², not just k - this is a common exam trap
Scale factor less than 1 produces a smaller figure (reduction)
To find the centre of enlargement, join corresponding vertices and extend the lines until they meet
All measurements (lengths, distances from centre) are multiplied by the scale factor in an enlargement

Enlargements (Leaving Cert Mathematics): Revision Notes

Enlargements

Key terminology

Scale factor

Drawing enlargements

When a vertex is the centre of enlargement

Enlargements with scale factor less than 1

Finding the centre of enlargement

Enlargement and area

Worked example: Finding scale factor and lengths

Worked example: Area problem

Inverse enlargements

Explore Leaving Cert Mathematics Model Answers by Topics

Enlargements and Constructions

Corollaries

Constructions

Transformations / Symmetries / Enlargements

Explore Leaving Cert Mathematics Quizzes by Topics

Enlargements and Constructions

Corollaries

Constructions

Transformations / Symmetries / Enlargements

Explore Leaving Cert Mathematics Flashcards by Topics

Enlargements and Constructions

Corollaries

Constructions

Transformations / Symmetries / Enlargements

Join 100,000+ Leaving Cert students studying Revision Notes with us.