Enlargement is a pivotal concept in geometry, involving the resizing of figures on a plane. This operation alters the dimensions of shapes but preserves their proportionality, governed by a scale factor and centred at a specific point. It's fundamental for students to grasp enlargement to understand spatial relationships and geometric transformations.

Image courtesy of Third Space Learning

Introduction to Enlargement

Enlargement, a transformation process, resizes shapes while maintaining their shape. It's defined by:

• Scale Factor (k): Dictates the altered size of the image relative to the original figure.
• Centre of Enlargement (C): The fixed point from which the transformation is measured.

Formula for Enlargement

For a point $P(x, y)$ enlarged from centre $C(a, b)$, the image $P'$ is given by:

Here, $P'$ is the image of $P$ after enlargement with scale factor $k$.

Impact of Scale Factor on Area

The area of a shape after enlargement changes by the square of the scale factor $(k^2)$, a vital concept illustrating the effect of dimensional changes on size.

Shape A has been enlarged by scale factor 2 to give shape B.

Worked Examples

Example 1: Basic Enlargement

Given a point P(2, 3) and a centre of enlargement C(1, 1) with a scale factor of 2, find the coordinates of the enlarged image $P'$.

Solution:

Using the enlargement formula:

$P'(1+2(2-1), 1+2(3-1))$

$P'(1+2(1), 1+4)$

$P'(3, 5)$

Therefore, the coordinates of the enlarged image $P'$ are (3, 5).

Example 2: Negative Scale Factor

Consider a point P(4, -2) with a centre of enlargement C(0, 0) and a scale factor of -1. Determine the coordinates of $P'$.

Solution:

Applying the formula:

$P'(0-1(4-0), 0-1(-2-0))$

$P'(-4, 2)$

The image $P'$ lies at (-4, 2), demonstrating how a negative scale factor results in an inversion along with enlargement.

Example 3: Area Change with Scale Factor

A rectangle has dimensions 5 units by 3 units. If it is enlarged by a scale factor of 3 from a centre of enlargement, calculate the area of the enlarged rectangle.

Solution:

Original Area: $5 \times 3 = 15$ square units

Scale Factor (k): 3

New Area: $15 \times 3^2 = 15 \times 9 = 135$ square units

The area of the rectangle increases by a factor of 9 (the square of the scale factor), resulting in a new area of 135 square units.

Practice Questions

Question 1:

Given: Triangle with vertices $A(1, 2)$, $B(4, 5)$, $C(6, 2)$; Centre of Enlargement at origin $(0, 0)$; Scale Factor $k = 2$.

Task: Find the coordinates of the new vertices after enlargement.

Solution:

The original triangle vertices are $A(1, 2)$, $B(4, 5)$, and $C(6, 2)$. After enlargement with a scale factor of 2 from the origin $(0, 0)$, the new vertices are:

• $A'(2, 4)$
• $B'(8, 10)$
• $C'(12, 4)$

The sketch below demonstrates the original triangle in blue and its enlarged version in red, showing how each point has moved away from the origin, doubling its distance from it.

Question 2:

Given: Square with side length 4 units; Scale Factor $k = 0.5.$

Task: Calculate the new area of the square after enlargement.

Solution:

1. Original Square Dimensions: Side length = 4 units.

2. Scale Factor Applied: $k = 0.5$

3. New Side Length Calculation:

• New side length = Original side length $\times$ Scale factor.
• New side length = $4 \times 0.5 = 2$ units.

4. Area Calculation:

• $\text{Original area}$ = $\text{Side length}^2$ = $4^2 = 16$ square units.
• $\text{New area}$ = $\text{New side length}^2$ = $2^2 = 4$ square units.

The sketch illustrates the original square in blue and its reduced version in red after applying the scale factor. Each side of the original square has been halved in length, leading to a new square that is a quarter the area of the original, aligning with our area calculations.

Tips for Mastery

• Draw Diagrams: Visual aids help comprehend enlargement effects.
• Experiment with Scale Factors: Practise with various $k$ values to observe size and position changes.
• Understand Area Changes: The area changes by $k^2$, crucial for quantifying enlargement effects.