Chapter 15: Congruence, Enlargement and Combined Transformations

Overview

Welcome to Chapter 15 of Form 5 Mathematics! This chapter explores the fascinating world of geometric transformations. You'll learn to distinguish between congruent and non-congruent shapes, understand enlargement transformations, and master the composition of multiple transformations. These concepts are fundamental to geometry, computer graphics, and many real-world applications.

What You'll Learn:

• Differentiate between congruent and non-congruent shapes
• Understand enlargement transformations and scale factors
• Describe combined transformations and solve related problems
• Apply transformation concepts to solve geometric problems

Learning Objectives

After completing this chapter, you will be able to:

• Differentiate between congruent and non-congruent shapes
• Explain the meaning of similarity and enlargement
• Determine the image and object of an enlargement
• Solve problems involving enlargement
• Describe combined transformations and solve problems involving them

Geometric Transformations Overview

Geometric transformations are operations that change the position, size, or shape of geometric figures. The four basic transformations are:

1. Translation - Sliding a figure without rotating or reflecting
2. Reflection - Flipping a figure over a line
3. Rotation - Turning a figure around a point
4. Enlargement - Scaling a figure by a factor

Key Concepts

Congruence

Congruence means two shapes have the same shape and size. Corresponding sides and angles are equal.

Properties of Congruent Shapes:

• Equal corresponding sides
• Equal corresponding angles
• Same shape and size

Example: Two triangles with all sides and angles equal are congruent.

Non-Congruent Shapes

Non-congruent shapes do not have both the same shape and size. They may differ in:

• Size (different dimensions)
• Shape (different angles or proportions)

Example: Two triangles with different side lengths or angle measures.

Similarity

Similarity means shapes have the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are proportional.

Properties of Similar Shapes:

• Equal corresponding angles
• Proportional corresponding sides

Enlargement

Enlargement is a transformation that changes the size of a shape but maintains its shape. It is described by:

• Center of enlargement (fixed point)
• Scale factor (k)

Properties of Enlargement:

• If |k| > 1, image is larger than object
• If 0 < |k| < 1, image is smaller than object
• If k > 0, image is on same side as object from center
• If k < 0, image is on opposite side and inverted

Combined Transformations

Combined transformations involve performing two or more transformations sequentially. The notation PQ means transformation Q is applied first, then transformation P.

Order Matters: Transformations are applied from right to left in notation.

Congruence Tests

Four Main Congruence Tests

Congruence Test Details

SSS (Side-Side-Side): If all three sides of one triangle are equal to all three sides of another triangle, then the triangles are congruent.

SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

RHS (Right-Hypotenuse-Side): If one triangle is a right-angled triangle with the hypotenuse and one side equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.

Congruence vs Similarity

Key Differences

Similarity Conditions

Enlargement Transformations

Properties of Enlargement

Scale Factor Effects

Coordinate Transformations

Enlargement from Origin (0,0):

$$(x', y') = (kx, ky)$$

Enlargement from Point (a,b):

$$(x', y') = (a + k(x-a), b + k(y-b))$$

Area Relationship:

$$\text{Area of Image} = k^2 \times \text{Area of Object}$$

Combined Transformations

Transformation Notation

Important Rule: In transformation notation AB, transformation B is applied first, then transformation A.

Order of Application: Right to left

Common Combined Transformations

Translation then Reflection:

Rotation then Enlargement:

Reflection then Translation then Rotation:

Transformation Composition Rules

• Translation + Translation = Translation
• Reflection + Reflection = Translation (if parallel lines)
• Rotation + Rotation = Rotation or Translation
• Enlargement + Enlargement = Enlargement
• Translation + Rotation = Rotation + Translation

Tessellation Patterns

Tessellation Requirements

Regular Tessellations

Semi-Regular Tessellations

Important Formulas and Methods

Enlargement Formulas

Scale Factor (k):

$$k = \frac{\text{Length of image side}}{\text{Length of object side}}$$

Area of Image under Enlargement:

$$\text{Area of Image} = k^2 \times \text{Area of Object}$$

Coordinates Transformation (Enlargement from origin): If center is (0,0) and scale factor is k:

$$(x', y') = (kx, ky)$$

If center is (a,b) and scale factor is k:

$$(x', y') = (a + k(x-a), b + k(y-b))$$

Tessellation

Tessellation is a repeating pattern of shapes that covers a plane without gaps or overlaps.

Requirements for Tessellation:

• Shapes fit together perfectly
• No gaps between shapes
• No overlapping of shapes
• Pattern repeats infinitely

Step-by-Step Solved Examples

Example 1: Congruence Tests

Problem: Determine which of the following pairs of triangles are congruent: a) Triangle ABC with AB = 5, BC = 6, AC = 7 Triangle DEF with DE = 5, EF = 6, DF = 7

b) Triangle PQR with angles 45°, 60°, 75° Triangle XYZ with angles 45°, 60°, 75° but sides twice as long

Solution: a) ABC and DEF: All three corresponding sides are equal (5, 6, 7). By SSS (Side-Side-Side) congruence test, the triangles are congruent.

b) PQR and XYZ: All corresponding angles are equal, but sides are different lengths. The triangles are similar but not congruent because they have different sizes.

Answer: a) Congruent, b) Not congruent

Example 2: Enlargement Scale Factor

Problem: A triangle has sides 3 cm, 4 cm, 5 cm. After enlargement, the corresponding sides are 9 cm, 12 cm, 15 cm. Find: a) The scale factor b) The center of enlargement if (2, 1) maps to (8, 5)

Solution: a) Scale Factor: k = 9/3 = 12/4 = 15/5 = 3

b) Center of Enlargement: Using (2, 1) → (8, 5) with k = 3 Let center be (a, b): 8 = a + 3(2 - a) = a + 6 - 3a = 6 - 2a 8 = 6 - 2a → 2a = 6 - 8 → 2a = -2 → a = -1

5 = b + 3(1 - b) = b + 3 - 3b = 3 - 2b 5 = 3 - 2b → 2b = 3 - 5 → 2b = -2 → b = -1

Answer: a) Scale factor = 3, b) Center = (-1, -1)

Example 3: Area after Enlargement

Problem: A rectangle with area 20 c$m^2$ is enlarged with scale factor 2.5. Find the area of the image.

Solution: Area of Image = $k^2$ × Area of Object Area of Image = (2.5)² × 20 Area of Image = 6.25 × 20 Area of Image = 125 c$m^2$

Answer: Area of image is 125 c$m^2$

Example 4: Combined Transformations

Problem: A shape undergoes transformations AB, where:

• A: Translation by vector (3, 2)
• B: Reflection over the line y = x

If point (1, 4) is the original position, find its final position.

Solution: Step 1: Apply B first (reflection over y = x) Reflection over y = x swaps x and y coordinates: (1, 4) → (4, 1)

Step 2: Apply A (translation by (3, 2)) Add translation vector: (4 + 3, 1 + 2) = (7, 3)

Answer: Final position is (7, 3)

Example 5: Real-world Application - Map Scaling

Problem: A map is drawn with scale 1:50,000. A rectangular park on the map measures 4 cm by 6 cm. The map is enlarged by a factor of 2 to make a poster.

Solution: Step 1: Find actual dimensions of park: Map scale: 1 cm = 50,000 cm = 500 m Actual length = 6 cm × 500 = 3,000 m Actual width = 4 cm × 500 = 2,000 m

Step 2: Find poster dimensions: Enlargement factor = 2 Poster length = 6 cm × 2 = 12 cm Poster width = 4 cm × 2 = 8 cm

Step 3: Find poster scale: 1 cm on poster = 1 cm / 2 = 0.5 cm on map 0.5 cm on map = 0.5 × 500 m = 250 m Poster scale: 1 cm = 250 m

Answer: Poster is 12 cm by 8 cm with scale 1:25,000

Real-world Applications

1. Architecture and Engineering

• Blueprint Scaling: Creating different-sized versions of plans
• Model Building: Scaling down designs for models
• Structural Analysis: Testing scaled models of structures

2. Computer Graphics

• Image Scaling: Resizing digital images
• 3D Modeling: Transforming 3D objects
• Animation: Creating smooth transformation sequences

3. Cartography and Maps

• Map Scales: Different zoom levels and representations
• GIS Systems: Geographic information transformations
• Navigation: Route planning and scaling

4. Manufacturing and Production

• Prototyping: Scaling designs for different production runs
• Quality Control: Ensuring dimensional accuracy
• Packaging Design: Scaling for different product sizes

Important Terms

Summary Points

• Congruent Shapes: Same shape and size, equal sides and angles
• Similar Shapes: Same shape, proportional sides, equal angles
• Enlargement: Changes size, maintains shape
• Scale Factor: k = image/object dimensions
• Area Scaling: Area Image = $k^2$ × Area Object
• Combined Transformations: Apply right to left (Q then P for PQ)
• Tessellation: Repeating pattern covering plane without gaps
• Enlargement Center: Fixed point reference for transformation

Practice Tips for SPM Students

1. Master Congruence Tests

• Learn SSS, SAS, ASA, AAS, RHS congruence tests
• Practice identifying congruent shapes
• Understand the difference between congruence and similarity

2. Enlargement Calculations

• Practice scale factor calculations
• Master area scaling formulas
• Learn coordinate transformation methods
• Understand center of enlargement determination

3. Combined Transformations

• Practice applying transformations in correct order
• Understand transformation notation
• Master coordinate calculations for combined operations

4. Common Mistakes to Avoid

• Confusing congruence with similarity
• Incorrect scale factor applications
• Wrong transformation order in combined operations
• Calculation errors in area scaling

SPM Exam Tips

Paper 1 (Multiple Choice)

• Look for congruence indicators (equal sides/angles)
• Remember scale factor relationships
• Practice quick transformation identification
• Use elimination method for difficult questions

Paper 2 (Structured)

• Show all transformation steps clearly
• Demonstrate scale factor calculations
• Explain congruence tests used
• Label centers of enlargement properly

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Did You Know? The concept of geometric transformations has been studied since ancient times, but it was during the Renaissance that artists like Leonardo da Vinci and Albrecht Dürer systematically studied perspective transformations. Today, transformations are fundamental in computer graphics, enabling everything from video games to special effects!

Next Chapter: In Chapter 6, you'll explore trigonometric ratios and graphs, learning to solve problems involving sine, cosine, and tangent functions and their graphs.

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• Chapter 4: Consumer Mathematics (Taxation)
• Chapter 6: Trigonometric Functions and Graphs