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
By the end of this lesson, you will be able to determine whether two geometric figures are similar, calculate the linear scale factor \(k\), and understand the exact mathematical relationship governing corresponding lengths, perimeters, and areas.
Core Conceptual Foundation
Two figures are defined as mathematically similar if and only if:
- All corresponding angles are equal (shape is preserved).
- All pairs of corresponding lengths are in the same constant ratio, known as the linear scale factor \(k\).
Imagine blueprinting a shamba (farm plot) in Nakuru. If the blueprint side is \(5\text{ cm}\) and the actual plot side is \(15\text{ m}\) (\(1500\text{ cm}\)), the linear scale factor is \(k = \frac{1500}{5} = 300\). Every single length on the actual plot is multiplied by \(300\), while angles remain unchanged.
Key Formulas
1. Linear Scale Factor (\(k\))
\[ k = \frac{\text{Dimension on Image (Enlarged / Reduced)}}{\text{Corresponding Dimension on Original Object}} \]2. Side Length & Perimeter Scaling
\[ \text{Image Length} = k \times \text{Original Length} \]\[ \frac{\text{Image Perimeter}}{\text{Original Perimeter}} = k \]3. Area Scale Factor (\(k^2\))
\[ \frac{\text{Area of Image}}{\text{Area of Original}} = k^2 \implies \text{Area of Image} = k^2 \times \text{Original Area} \]4. Proportionality of Corresponding Sides
For any two similar polygons with sides \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\):
\[ \frac{a_2}{a_1} = \frac{b_2}{b_1} = \frac{c_2}{c_1} = k \]Worked Examples
Example 1 (Easy): Finding Dimensions Using Linear Scale Factor
A triangular piece of Maasai Shuka has sides of length \(6\text{ cm}\), \(8\text{ cm}\), and \(10\text{ cm}\). An artisan creates an enlarged similar pattern with a scale factor of \(k = 3.5\). Find the lengths of all three sides of the enlarged pattern.
- Identify the linear scale factor: \(k = 3.5\).
- Multiply each side by \(k\): \[ \text{Side}_1 = 6 \times 3.5 = 21\text{ cm} \] \[ \text{Side}_2 = 8 \times 3.5 = 28\text{ cm} \] \[ \text{Side}_3 = 10 \times 3.5 = 35\text{ cm} \]
- Conclusion: The sides of the enlarged triangular pattern are \(21\text{ cm}\), \(28\text{ cm}\), and \(35\text{ cm}\).
Example 2 (Medium): Calculating Area from Linear Dimensions
A rectangular community garden in Machakos measures \(4\text{ m}\) by \(6\text{ m}\). The county plans to build an expanded similar garden with a length of \(18\text{ m}\) corresponding to the \(6\text{ m}\) side. Find the area of the expanded garden.
- Calculate original area: \[ \text{Area}_{\text{orig}} = 4\text{ m} \times 6\text{ m} = 24\text{ m}^2 \]
- Determine the linear scale factor \(k\): \[ k = \frac{\text{Image Length}}{\text{Original Length}} = \frac{18}{6} = 3 \]
- Apply the Area Scale Factor (\(k^2\)): \[ \text{Area Scale Factor} = k^2 = 3^2 = 9 \]
- Calculate the enlarged area: \[ \text{Area}_{\text{expanded}} = k^2 \times \text{Area}_{\text{orig}} = 9 \times 24 = 216\text{ m}^2 \]
- Check using new dimensions: New width \(= 4 \times 3 = 12\text{ m}\). Area \(= 12 \times 18 = 216\text{ m}^2\). Both methods agree!
Example 3 (Hard): Finding Unknown Dimensions from an Area Ratio
A logo for a Kenyan tech startup has an original area of \(32\text{ cm}^2\) and a base of \(8\text{ cm}\). When printed on a promotional billboard, the logo covers an area of \(1\,800\text{ cm}^2\). Determine the linear scale factor and the base length of the billboard logo.
- Set up the area ratio formula: \[ \frac{\text{Area}_{\text{billboard}}}{\text{Area}_{\text{orig}}} = k^2 \implies k^2 = \frac{1800}{32} = \frac{225}{4} = 56.25 \]
- Solve for linear scale factor \(k\): \[ k = \sqrt{56.25} = 7.5 \quad \left(\text{or } \sqrt{\frac{225}{4}} = \frac{15}{2} = 7.5\right) \]
- Compute the new base length: \[ \text{New Base} = k \times \text{Original Base} = 7.5 \times 8 = 60\text{ cm} \]
- Conclusion: The linear scale factor is \(7.5\) and the billboard logo base is \(60\text{ cm}\).
Common Mistakes
Why it feels right: We intuitively expect everything to grow at the same rate.
Correction: Area has two dimensions (length and width). If both are scaled by \(k\), the area increases by \(k \times k = k^2\). For instance, doubling sides (\(k=2\)) quadruples the area (\(k^2=4\)).
Correction: In similar shapes, corresponding angles remain exactly identical. If the angles change, the shape gets distorted and is no longer similar.
Correction: Always match the shortest side with the shortest side, or the side between corresponding equal angles. Dividing arbitrary sides leads to an incorrect scale factor.
Real World
A digital surveyor map of Kisumu uses a representative fraction of \(1 : 50\,000\). A distance of \(4\text{ cm}\) on the phone screen represents \(4 \times 50\,000 = 200\,000\text{ cm} = 2\text{ km}\) of actual road distance.
If a farmer triples (\(k = 3\)) the side lengths of a rectangular greenhouse, the required plastic roof and soil coverage increase by \(k^2 = 9\). Seedling and fertilizer budgets must scale by 9 times, not 3 times.
Structural engineers build miniature wind tunnel models of skyscrapers like Britam Tower to evaluate airflow and stability. Geometric similarity guarantees that aerodynamic forces modeled at scale accurately predict full-scale behaviour.
Practice