Similarity and Enlargement
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective
Understand the geometry of enlargement transformations using both positive and negative scale factors, determine centre of enlargements, and relate linear scale factor to perimeter and area changes.
Concrete Scenario: The Kitenge Fabric & Photocopier
Imagine a fashion designer in Gikomba market scaling up a traditional Kitenge pattern onto a massive billboard in Nairobi. If the original motif is doubled along every edge, the linear scale factor is \(k = 2\). If it is reduced to half-size for a pocket square, \(k = 0.5\). In geometric enlargements, all corresponding angles remain strictly identical, while lengths scale proportionally by \(|k|\).
Interactive Enlargement Projector
Drag the yellow centre point \(O\). Adjust the scale factor \(k\) slider. Notice how negative values of \(k\) invert and project the image through the centre point!
Ray Projection & Coordinates
An enlargement from a fixed centre \(O\) maps any point \(P\) to \(P'\) along the ray \(OP\) such that: \[ \vec{OP'} = k \cdot \vec{OP} \] When \(k > 0\), the image \(P'\) lies on the same side of \(O\). When \(k < 0\), the vector is negated, projecting \(P'\) through \(O\) to the opposite side, naturally rotating the figure by \(180^\circ\).
Key Formulas
Linear Scale Factor (\(k\))
\[ k = \frac{\text{Image Length}}{\text{Original Length}} = \frac{A'B'}{AB} \]Vector Transformation Rule
\[ \vec{OP'} = k \times \vec{OP} \quad \Longleftrightarrow \quad P' = O + k(P - O) \]Area and Perimeter Scaling
\[ \text{New Perimeter} = |k| \times \text{Original Perimeter} \]\[ \text{New Area} = k^2 \times \text{Original Area} \]Volume Scale Factor (3D Objects)
\[ \text{New Volume} = |k|^3 \times \text{Original Volume} \]Worked Examples
Example 1 (Easy): Basic Side Length Enlargement
A rectangular plot of land on an estate map measures \(4\text{ cm}\) by \(6\text{ cm}\). The architect enlarges the drawing by a linear scale factor of \(k = 3\). Find the dimensions of the enlarged rectangle.
- Identify formula: \(\text{New Length} = k \times \text{Original Length}\).
- Calculate: \[ \text{Length} = 3 \times 4\text{ cm} = 12\text{ cm} \]\[ \text{Width} = 3 \times 6\text{ cm} = 18\text{ cm} \]
- Conclusion: The enlarged rectangle has dimensions \(12\text{ cm} \times 18\text{ cm}\).
Example 2 (Medium): Area Transformation
A triangular community garden in Kisumu has an area of \(30\text{ m}^2\). The youth group expands the garden uniformly using a linear scale factor of \(k = 2.5\). Calculate the area of the expanded garden.
- Recall the area scaling relationship: \[ \text{Area}_{\text{new}} = k^2 \times \text{Area}_{\text{original}} \]
- Compute \(k^2\): \[ k^2 = (2.5)^2 = 6.25 \]
- Calculate new area: \[ \text{Area}_{\text{new}} = 6.25 \times 30 = 187.5\text{ m}^2 \]
Example 3 (Hard): Negative Scale Factor & Distance from Centre
A point \(P\) is located \(6\text{ cm}\) from the centre of enlargement \(O\). The image point \(P'\) is projected to the opposite side of \(O\) such that \(OP' = 15\text{ cm}\). Determine the scale factor \(k\) and describe the transformation.
- Find magnitude of \(k\): \[ |k| = \frac{OP'}{OP} = \frac{15}{6} = 2.5 \]
- Determine sign: Because \(P'\) is on the opposite side of \(O\), the scale factor is negative: \[ k = -2.5 \]
- Geometric description: The figure is enlarged by a factor of \(2.5\) and rotated \(180^\circ\) about the centre \(O\).
Common Mistakes
Misconception 1: Confusing Linear Scale Factor with Area Scale Factor
Misconception 2: Thinking Negative Scale Factors Make Shapes "Smaller than Zero"
Misconception 3: Measuring from the Edge Instead of the Centre \(O\)
Real World
Practice