Solved Exercises on Scale Factors in Grade 7

Master scale factors: finding scale factors, calculating dimensions, comparing areas and perimeters of scaled figures through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Scale Factor
Exercise 1
Rectangle ABCD has length 6 cm and width 4 cm. Rectangle A'B'C'D' has length 12 cm and width 8 cm. Find the scale factor.
Definition:

Scale Factor: The ratio of corresponding lengths in two similar figures. Scale Factor = New Length ÷ Original Length

Method to find scale factor:
  1. Identify corresponding sides in both figures
  2. Divide the length of the scaled side by the original side
  3. Verify with another pair of corresponding sides
Original
Length: 6cm, Width: 4cm
Scaled
Length: 12cm, Width: 8cm
Scale Factor
2
Step 1: Identify corresponding sides

Original length = 6cm, Scaled length = 12cm

Original width = 4cm, Scaled width = 8cm

Step 2: Calculate scale factor using length

Scale Factor = Scaled Length ÷ Original Length

Scale Factor = 12 ÷ 6 = 2

Step 3: Verify with width

Scale Factor = Scaled Width ÷ Original Width

Scale Factor = 8 ÷ 4 = 2

Step 4: Confirm consistency

Both calculations give the same result: Scale Factor = 2

Scale Factor = 2
Final answer:

The scale factor from rectangle ABCD to A'B'C'D' is 2.

Applied rules:

• Scale Factor Formula: SF = New Length ÷ Original Length

• Consistency Check: All corresponding sides must have the same scale factor

• Similar Figures: Corresponding angles remain equal

2 Calculating Scaled Dimensions
Exercise 2
Triangle PQR has sides of length 3 cm, 4 cm, and 5 cm. If the triangle is enlarged by a scale factor of 3, find the lengths of the sides of the new triangle P'Q'R'.
Definition:

Scaling Up: When the scale factor is greater than 1, the figure becomes larger. New Length = Original Length × Scale Factor

Original
3cm, 4cm, 5cm
Scale Factor
3
Scaled
9cm, 12cm, 15cm
Step 1: List original side lengths

PQ = 3cm, QR = 4cm, PR = 5cm

Step 2: Apply scaling formula to each side

New Length = Original Length × Scale Factor

P'Q' = 3 × 3 = 9cm

Q'R' = 4 × 3 = 12cm

P'R' = 5 × 3 = 15cm

Step 3: Verify the scale factor

P'Q'/PQ = 9/3 = 3 ✓

Q'R'/QR = 12/4 = 3 ✓

P'R'/PR = 15/5 = 3 ✓

P'Q' = 9cm, Q'R' = 12cm, P'R' = 15cm
Final answer:

The sides of triangle P'Q'R' are 9cm, 12cm, and 15cm.

Applied rules:

• Scaling Formula: New Length = Original Length × Scale Factor

• Uniform Scaling: All sides scale by the same factor

• Shape Preservation: Angles remain unchanged in scaled figures

3 Finding Original Dimensions
Exercise 3
Rectangle EFGH is a scaled copy of rectangle ABCD. The scale factor from ABCD to EFGH is 0.5. If EFGH has dimensions 7 cm by 5 cm, find the dimensions of ABCD.
Definition:

Scaling Down: When the scale factor is less than 1, the figure becomes smaller. Original Length = Scaled Length ÷ Scale Factor

Scaled
7cm × 5cm
Scale Factor
0.5
Original
14cm × 10cm
Step 1: Identify given information

Scale Factor = 0.5 (scaling down)

Scaled dimensions: Length = 7cm, Width = 5cm

Step 2: Apply reverse scaling formula

Original Length = Scaled Length ÷ Scale Factor

Original Length = 7 ÷ 0.5 = 14cm

Original Width = 5 ÷ 0.5 = 10cm

Step 3: Verify the calculation

14 × 0.5 = 7cm ✓

10 × 0.5 = 5cm ✓

Step 4: Present the answer

Original rectangle ABCD has dimensions 14cm by 10cm

ABCD: 14cm × 10cm
Final answer:

Rectangle ABCD has dimensions 14cm by 10cm.

Applied rules:

• Reverse Scaling: Original = Scaled ÷ Scale Factor

• Fractional Scale Factors: Values less than 1 create smaller copies

• Division by Decimal: Dividing by 0.5 equals multiplying by 2

Scale Factor Rules and Methods
\(SF = \frac{\text{New Length}}{\text{Original Length}}\)
Scale Factor Formula
Scale Factor > 1
Enlargement
Figure becomes larger
Scale Factor = 1
Congruent
Same size and shape
Scale Factor < 1
Reduction
Figure becomes smaller
Key definitions:

Scale Factor: The ratio of corresponding lengths in similar figures

Similar Figures: Figures with same shape but possibly different sizes

Corresponding Parts: Matching sides and angles in similar figures

Complete methodology:
  1. Identify Similarity: Confirm figures are similar (same shape)
  2. Find Corresponding Parts: Match sides and vertices
  3. Calculate Scale Factor: Use SF = New/Original
  4. Apply Scale Factor: Find missing dimensions
Tip 1: Always check that all corresponding sides give the same scale factor.
Tip 2: When scale factor is a fraction, divide by the reciprocal.
Tip 3: Areas scale by the square of the scale factor.
Tip 4: Perimeters scale by the same factor as lengths.
Common errors: Mixing up original and scaled measurements, forgetting to apply scale factor to all dimensions.
Exam preparation: Practice with various scale factors, including decimals and fractions.
Formulas to know by heart:

• Scale Factor = New Length ÷ Original Length

• New Length = Original Length × Scale Factor

• Original Length = New Length ÷ Scale Factor

• Scaled Area = Original Area × (Scale Factor)²

• Scaled Perimeter = Original Perimeter × Scale Factor

Solution: Exercises 4 to 5
4 Area and Perimeter Scaling
Exercise 4
Square PQRS has side length 4 cm. It is enlarged by a scale factor of 2.5. Find the area and perimeter of both squares.
Definition:

Area Scaling: Area scales by the square of the scale factor. Perimeter scales by the scale factor itself.

Original
Side: 4cm
Scale Factor
2.5
Scaled
Side: 10cm
Step 1: Calculate original area and perimeter

Original Area = Side² = 4² = 16 cm²

Original Perimeter = 4 × Side = 4 × 4 = 16 cm

Step 2: Find scaled side length

Scaled Side = Original Side × Scale Factor

Scaled Side = 4 × 2.5 = 10 cm

Step 3: Calculate scaled area and perimeter

Scaled Area = Side² = 10² = 100 cm²

Scaled Perimeter = 4 × Side = 4 × 10 = 40 cm

Step 4: Verify scaling relationships

Area Ratio = 100/16 = 6.25 = (2.5)² ✓

Perimeter Ratio = 40/16 = 2.5 ✓

Original: Area = 16cm², Perimeter = 16cm
Scaled: Area = 100cm², Perimeter = 40cm
Final answer:

Original square: Area = 16cm², Perimeter = 16cm

Scaled square: Area = 100cm², Perimeter = 40cm

Applied rules:

• Area Scaling: Area changes by (Scale Factor)²

• Perimeter Scaling: Perimeter changes by Scale Factor

• Dimension Analysis: Lengths scale by SF, areas by SF²

5 Complex Scaling Problem
Exercise 5
Triangle XYZ has sides 5 cm, 12 cm, and 13 cm. A scaled copy of this triangle has its longest side measuring 26 cm. Find the scale factor and the lengths of the other two sides of the scaled triangle.
Definition:

Proportional Scaling: In similar triangles, all corresponding sides are proportional by the same scale factor.

Original
5, 12, 13 cm
Scale Factor
2
Scaled
10, 24, 26 cm
Step 1: Identify corresponding parts

Original longest side = 13 cm

Scaled longest side = 26 cm

Step 2: Calculate scale factor

Scale Factor = Scaled Longest Side ÷ Original Longest Side

Scale Factor = 26 ÷ 13 = 2

Step 3: Apply scale factor to other sides

Scaled Shortest Side = 5 × 2 = 10 cm

Scaled Middle Side = 12 × 2 = 24 cm

Step 4: Verify the results

Check ratios: 10/5 = 2, 24/12 = 2, 26/13 = 2 ✓

All ratios equal the scale factor

Scale Factor = 2, Scaled sides: 10cm, 24cm, 26cm
Final answer:

Scale Factor = 2, Other sides: 10cm and 24cm

Applied rules:

• Proportionality: All corresponding sides scale by the same factor

• Triangle Properties: Right triangle (5-12-13 is Pythagorean triple)

• Verification: Always check that all sides follow the same scale factor

Scale Factor Laws, Methods, Rules, and Definitions
\(SF = \frac{\text{Scaled Length}}{\text{Original Length}}\)
Scale Factor
Key definitions:

Scale Factor: The constant multiplier that relates corresponding lengths in similar figures

Similar Figures: Geometric figures that have the same shape but different sizes

Corresponding Parts: Matching sides, angles, or vertices in similar figures

Complete methodology:
  1. Identify Similarity: Confirm figures are similar (same shape, proportional sides)
  2. Find Corresponding Parts: Match sides and vertices in both figures
  3. Calculate Scale Factor: Divide scaled measurement by original measurement
  4. Apply Scale Factor: Use to find unknown measurements
  5. Verify Results: Check that all corresponding parts follow the same scale factor
Tip 1: Always identify the longest and shortest sides first when matching corresponding parts.
Tip 2: If scale factor is a decimal, convert to fraction to make calculations easier.
Tip 3: For area problems, remember to square the scale factor.
Tip 4: For volume problems (Grade 8+), cube the scale factor.
Common errors: Mixing up original and scaled measurements, forgetting to square the scale factor for areas, not checking all corresponding parts.
Exam preparation: Practice with various geometric shapes, work with decimals and fractions, solve multi-step problems.
Formulas to know by heart:

• Scale Factor = Scaled Length ÷ Original Length

• Scaled Length = Original Length × Scale Factor

• Original Length = Scaled Length ÷ Scale Factor

• Scaled Area = Original Area × (Scale Factor)²

• Scaled Perimeter = Original Perimeter × Scale Factor

• Scaled Volume = Original Volume × (Scale Factor)³

Visualizing Scale Factors: Geometric Transformations
Exercise 6: Scale Factor Relationships
Consider geometric figures with different scale factors:
Original square (side = 2cm), Scale Factor = 1.5
Original rectangle (3cm×4cm), Scale Factor = 2
Original triangle (base=4cm, height=3cm), Scale Factor = 0.5

Analysis: The chart shows how dimensions, perimeters, and areas change with different scale factors.

  • Linear dimensions scale by the scale factor
  • Areas scale by the square of the scale factor
  • Scale factors greater than 1 enlarge figures
  • Scale factors less than 1 reduce figures

Questions & Answers

Question: I'm confused about why areas scale by the square of the scale factor. Can you explain this concept?

Answer: Great question! The reason areas scale by the square of the scale factor is because area is a two-dimensional measurement:

  • Area = length × width (two dimensions)
  • If both length and width are multiplied by the scale factor (SF), then:
  • New Area = (SF × length) × (SF × width) = SF² × (length × width)

Example: If you have a rectangle 3cm × 4cm (area = 12cm²) and scale it by factor 2:

  • New dimensions: (2×3cm) × (2×4cm) = 6cm × 8cm
  • New area: 6cm × 8cm = 48cm²
  • Notice: 48cm² = 4 × 12cm² = (2²) × 12cm²

This is why area always scales by SF² and volume by SF³!

Question: When I have a scale factor of 0.5, does that mean the new figure is half the size? It seems counterintuitive.

Answer: Yes, a scale factor of 0.5 means the new figure is half the size of the original! Here's how to think about it:

  • Scale factor of 0.5 = 1/2
  • New length = Original length × 0.5 = Original length ÷ 2
  • This makes the new figure half the size of the original

Think of it as: "multiply by 0.5 to get half the size" or "divide by 2".

Examples:

  • Original: 10cm, Scale Factor: 0.5 → New: 10 × 0.5 = 5cm
  • Original: 8cm, Scale Factor: 0.25 → New: 8 × 0.25 = 2cm

Fractional scale factors (less than 1) always create smaller copies!

Question: How do I know if two figures are similar? What should I look for?

Answer: Two figures are similar if they meet these criteria:

  1. Equal Corresponding Angles: All matching angles are equal
  2. Proportional Corresponding Sides: All matching sides have the same scale factor

How to check:

  • Measure all angles - they should match between figures
  • Compare ratios of corresponding sides - they should all be equal
  • Look for the same shape but possibly different sizes

Examples of similar figures: all circles, all squares, all equilateral triangles

Note: Congruent figures are also similar (scale factor = 1)