Solved Exercises on Scale Factors in Grade 8

Master scale factors: enlargement, reduction, map scales, and geometric relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Enlargement with Scale Factor
Exercise 1
Rectangle ABCD has dimensions 4 cm × 6 cm. Find the dimensions of the enlarged rectangle A'B'C'D' if the scale factor is 2.5.
Definition:

Scale Factor: The ratio of the size of an image to the size of the original figure

Enlargement: When the scale factor is greater than 1, the figure becomes larger

Reduction: When the scale factor is between 0 and 1, the figure becomes smaller

Scale Factor Application Method:
  1. Identify the original dimensions
  2. Identify the scale factor
  3. Multiply each dimension by the scale factor
  4. Calculate the new dimensions
  5. Verify the result makes sense
Original Dimensions
4 cm × 6 cm
Scale Factor
k = 2.5
New Dimensions
10 cm × 15 cm
Step 1: Identify Original Dimensions

Original rectangle has length = 6 cm and width = 4 cm

Step 2: Identify Scale Factor

Scale factor k = 2.5 (enlargement since k > 1)

Step 3: Apply Scale Factor to Length

New length = Original length × Scale factor

New length = 6 × 2.5 = 15 cm

Step 4: Apply Scale Factor to Width

New width = Original width × Scale factor

New width = 4 × 2.5 = 10 cm

Step 5: Verify Result

Since 2.5 > 1, the new rectangle should be larger ✓

New dimensions: 10 cm × 15 cm

A'B'C'D' has dimensions 10 cm × 15 cm
Final answer:

The enlarged rectangle A'B'C'D' has dimensions 10 cm × 15 cm

Applied rules:

• Scale Factor Formula: New dimension = Original dimension × Scale factor

• Enlargement: Scale factor > 1

• Reduction: 0 < Scale factor < 1

2 Reduction and Finding Scale Factor
Exercise 2
Triangle PQR has sides of length 8 cm, 12 cm, and 16 cm. A similar triangle P'Q'R' has corresponding sides of 4 cm, 6 cm, and 8 cm. Find the scale factor and determine if this is an enlargement or reduction.
Definition:

Scale Factor Formula: Scale factor = New dimension / Original dimension

Similar Figures: Figures with the same shape but possibly different sizes

Proportional Sides: Corresponding sides of similar figures are in the same ratio

Original Triangle
8, 12, 16 cm
New Triangle
4, 6, 8 cm
Scale Factor
k = 1/2
Step 1: Identify Corresponding Sides

Original sides: 8 cm, 12 cm, 16 cm

Corresponding new sides: 4 cm, 6 cm, 8 cm

Step 2: Calculate Scale Factor

Scale factor = New side / Original side

Using first pair: k = 4/8 = 1/2

Using second pair: k = 6/12 = 1/2

Using third pair: k = 8/16 = 1/2

Step 3: Verify Consistency

All ratios equal 1/2, confirming the scale factor

Step 4: Determine Type

Since k = 1/2 = 0.5 < 1, this is a reduction

Step 5: Verify Result

Each new dimension is half the original, confirming the reduction

Scale factor = 1/2, this is a reduction
Final answer:

The scale factor is 1/2, and this represents a reduction

Applied rules:

• Scale Factor: k = New dimension / Original dimension

• Enlargement: k > 1

• Reduction: 0 < k < 1

• Consistency: All corresponding sides must have the same ratio

3 Map Scale Problems
Exercise 3
A map has a scale of 1:25,000. If the distance between two towns on the map is 6.4 cm, what is the actual distance between the towns in kilometers?
Definition:

Map Scale: A ratio showing how distances on the map relate to actual distances in reality

Scale Ratio: Written as 1:n, meaning 1 unit on the map represents n units in reality

Proportional Reasoning: Using ratios to solve real-world problems

Map Scale
1:25,000
Map Distance
6.4 cm
Actual Distance
1.6 km
Step 1: Understand the Scale

Scale 1:25,000 means 1 cm on the map represents 25,000 cm in reality

Step 2: Set Up Proportion

Map distance / Actual distance = Scale factor

6.4 cm / Actual distance = 1 / 25,000

Step 3: Solve for Actual Distance

Actual distance = 6.4 × 25,000 = 160,000 cm

Step 4: Convert to Kilometers

160,000 cm = 1,600 m = 1.6 km

Step 5: Verify Reasonableness

6.4 cm on map representing 1.6 km is reasonable for a town-to-town distance

Actual distance = 1.6 km
Final answer:

The actual distance between the towns is 1.6 km

Applied rules:

• Map Scale: 1:n means 1 unit maps to n units in reality

• Proportion: Map distance / Actual distance = Scale ratio

• Unit Conversion: Convert to appropriate units for the answer

Rules and methods, laws,...
\(k = \frac{\text{new dimension}}{\text{original dimension}}\)
Scale Factor Formula
Enlargement
\(k > 1\)
Figure gets larger
Reduction
\(0 < k < 1\)
Figure gets smaller
Area Ratio
\(k^2\)
Area scales by k²
Key definitions:

Similar Figures: Figures with the same shape but different sizes

Corresponding Parts: Parts that have the same relative position in similar figures

Proportional: Having the same ratio between corresponding parts

Dilation: A transformation that changes size by a scale factor

Complete methodology:
  1. Identify Given Information: Original dimensions, scale factor, or map scale
  2. Determine Required Calculation: Find new dimensions, scale factor, or actual distances
  3. Select Appropriate Formula: Scale factor, proportion, or conversion formula
  4. Substitute Known Values: Plug measurements into formula
  5. Solve Step-by-Step: Follow mathematical operations carefully
  6. Convert Units: Ensure answer is in correct units
  7. Verify Solution: Check if answer makes sense in context
Tip 1: Scale factor > 1 means enlargement, 0 < scale factor < 1 means reduction.
Tip 2: For areas, use k²; for volumes, use k³.
Tip 3: Always check that corresponding sides have the same scale factor.
Tip 4: In map scales, 1:n means 1 unit on map = n units in reality.
Common errors: Mixing up original and new dimensions, forgetting to convert units, inconsistent scale factors.
Exam preparation: Practice with various scale factors, understand area/volume relationships, work with different units.
Formulas to know by heart:

• Scale Factor: k = new dimension / original dimension

• Enlargement: k > 1

• Reduction: 0 < k < 1

• Area Ratio: k²

• Volume Ratio: k³

• Perimeter Ratio: k

Exercise with Visualization: Scale Factor Effects
Exercise 6: Scale Factor and Area Relationship
Compare how scale factors affect different properties:
Original square: side 2 cm, area 4 cm², perimeter 8 cm
Scaled square: side 6 cm, area 36 cm², perimeter 24 cm
Show the relationship between scale factors and properties.

Analysis: The chart shows how scale factors affect different properties.

  • Linear scale factor: 6/2 = 3
  • Perimeter scale factor: 24/8 = 3 (same as linear)
  • Area scale factor: 36/4 = 9 (3²)
  • Volume scale factor: 3³ = 27 (for 3D figures)

Questions & Answers

Question: How can I tell if a scale factor represents an enlargement or a reduction?

Answer: The scale factor determines the type of transformation:

  • Enlargement: Scale factor > 1 (figure becomes larger)
  • Reduction: 0 < Scale factor < 1 (figure becomes smaller)
  • No Change: Scale factor = 1 (figure stays the same size)
  • Reflection & Enlargement: Scale factor < 0 (includes reflection)

Examples:

  • k = 2, 3, 1.5 → Enlargements
  • k = 0.5, 0.25, 0.75 → Reductions
  • k = 1 → No change in size
  • k = -2 → Enlargement with reflection

The absolute value of the scale factor determines the size change, while the sign determines if there's also a reflection.

If |k| > 1, it's an enlargement; if 0 < |k| < 1, it's a reduction.

Question: How do scale factors affect area and volume? Is it the same as linear dimensions?

Answer: Scale factors affect different measurements differently:

  • Linear Dimensions: Scale factor k (directly proportional)
  • Areas: Scale factor k² (squared)
  • Volumes: Scale factor k³ (cubed)

For example, if the scale factor is 3:

  • Linear measurements (sides, heights): increase by factor of 3
  • Areas: increase by factor of 3² = 9
  • Volumes: increase by factor of 3³ = 27

This happens because:

  • Area involves 2 dimensions (length × width)
  • Volume involves 3 dimensions (length × width × height)

So if each dimension is multiplied by k, then area is multiplied by k² and volume by k³.

This is why a 2D shape becomes 4 times larger when scaled by 2, not 2 times larger!

Question: What properties of a figure stay the same when it's scaled? What changes?

Answer: When a figure is scaled by a scale factor, the following properties STAY THE SAME:

  • Shape of the figure
  • All angle measures
  • Parallelism (parallel lines remain parallel)
  • Perpendicularity (perpendicular lines remain perpendicular)
  • Ratio of corresponding linear measurements
  • Basic geometric relationships

What CHANGES:

  • Size of the figure (all lengths change by the scale factor)
  • Distance from any point to a reference point
  • Perimeter (changes by scale factor)
  • Area (changes by square of scale factor)
  • Volume (changes by cube of scale factor)

This is why similar figures have the same shape but different sizes. The geometric relationships remain consistent while the absolute measurements change proportionally.

For example, if you have two similar triangles, their corresponding angles are identical, but one triangle's sides are all longer or shorter than the other's by the same factor.

Question: Where are scale factors used in real life?

Answer: Scale factors have many practical applications:

  1. Maps and Blueprints: Creating scaled representations of real objects
  2. Photography: Zooming, resizing, and scaling images
  3. Architecture: Building models and creating construction plans
  4. Engineering: Designing parts that need to be scaled up or down
  5. Medicine: Enlarging medical images for diagnosis
  6. Cartography: Creating maps at different scales
  7. Computer Graphics: Resizing and scaling objects in design software

When you see a map with a scale like "1 inch = 10 miles," that's a practical application of scale factors! The mathematical concept models this common real-world scaling process.

In manufacturing, parts are often designed at one scale and then produced at a different scale using scale factor principles.

Question: How do I convert between map distance and actual distance using scale?

Answer: Use the scale ratio to convert between map and actual distances:

If the scale is 1:n:

  • Actual distance = Map distance × n
  • Map distance = Actual distance ÷ n

For example, with scale 1:50,000:

  • If map distance is 3 cm, actual distance = 3 × 50,000 = 150,000 cm = 1.5 km
  • If actual distance is 10 km, map distance = 10,000,000 ÷ 50,000 = 200 cm

Steps to solve:

  1. Identify the scale (e.g., 1:25,000)
  2. Identify what you're looking for (map or actual distance)
  3. Set up the proportion: (map distance)/(actual distance) = 1/n
  4. Solve for the unknown
  5. Convert units if necessary

Always pay attention to units and convert them appropriately!