Solved Exercises on Scaling Figures Up and Down in Grade 7

Master scaling figures: enlargements, reductions, and transformations using scale factors through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Enlargement
Exercise 1
Rectangle ABCD has dimensions 4 cm by 6 cm. It is enlarged by a scale factor of 2. Find the dimensions and area of the new rectangle A'B'C'D'.
Definition:

Enlargement: A transformation that increases the size of a figure. When the scale factor is greater than 1, the figure becomes larger.

Method for enlargement:
  1. Multiply each dimension by the scale factor
  2. Calculate the new area using the new dimensions
  3. Verify that area scales by (scale factor)²
Original
4cm × 6cm
Scale Factor
2
New
8cm × 12cm
Step 1: Identify original dimensions

Length = 6 cm, Width = 4 cm

Step 2: Apply scale factor to each dimension

New Length = Original Length × Scale Factor

New Length = 6 × 2 = 12 cm

New Width = Original Width × Scale Factor

New Width = 4 × 2 = 8 cm

Step 3: Calculate original and new areas

Original Area = 4 × 6 = 24 cm²

New Area = 8 × 12 = 96 cm²

Step 4: Verify area relationship

New Area = Original Area × (Scale Factor)²

96 = 24 × 2² = 24 × 4 = 96 ✓

A'B'C'D': 8cm × 12cm, Area = 96cm²
Final answer:

The new rectangle A'B'C'D' has dimensions 8 cm by 12 cm and an area of 96 cm².

Applied rules:

• Enlargement Formula: New Dimension = Original Dimension × Scale Factor

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

• Linear Scaling: Lengths change by Scale Factor

2 Reduction
Exercise 2
Triangle PQR has sides of length 10 cm, 8 cm, and 6 cm. It is reduced by a scale factor of 0.5. Find the lengths of the sides of the new triangle P'Q'R'.
Definition:

Reduction: A transformation that decreases the size of a figure. When the scale factor is between 0 and 1, the figure becomes smaller.

Original
10, 8, 6 cm
Scale Factor
0.5
New
5, 4, 3 cm
Step 1: Identify original side lengths

PQ = 10 cm, QR = 8 cm, RP = 6 cm

Step 2: Apply scale factor to each side

P'Q' = PQ × Scale Factor = 10 × 0.5 = 5 cm

Q'R' = QR × Scale Factor = 8 × 0.5 = 4 cm

R'P' = RP × Scale Factor = 6 × 0.5 = 3 cm

Step 3: Verify the scale factor consistency

P'Q'/PQ = 5/10 = 0.5 ✓

Q'R'/QR = 4/8 = 0.5 ✓

R'P'/RP = 3/6 = 0.5 ✓

Step 4: Calculate perimeters

Original Perimeter = 10 + 8 + 6 = 24 cm

New Perimeter = 5 + 4 + 3 = 12 cm

Check: 12/24 = 0.5 ✓

P'Q' = 5 cm, Q'R' = 4 cm, R'P' = 3 cm
Final answer:

The sides of triangle P'Q'R' are 5 cm, 4 cm, and 3 cm.

Applied rules:

• Reduction Formula: New Dimension = Original Dimension × Scale Factor

• Perimeter Scaling: Perimeter changes by Scale Factor

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

3 Comparing Enlargement and Reduction
Exercise 3
Square WXYZ has side length 5 cm. Find the dimensions of: (a) an enlargement with scale factor 3, and (b) a reduction with scale factor 0.4.
Definition:

Scale Factor Comparison: Scale factors greater than 1 cause enlargement, while scale factors between 0 and 1 cause reduction.

Original
5 cm side
Enlargement
SF=3 → 15cm
Reduction
SF=0.4 → 2cm
Step 1: Calculate enlargement (SF = 3)

New Side Length = Original Side × Scale Factor

New Side Length = 5 × 3 = 15 cm

Step 2: Calculate reduction (SF = 0.4)

New Side Length = Original Side × Scale Factor

New Side Length = 5 × 0.4 = 2 cm

Step 3: Compare the results

Original: 5 cm → Enlargement: 15 cm (3× larger)

Original: 5 cm → Reduction: 2 cm (2.5× smaller)

Step 4: Calculate areas for comparison

Original Area = 5² = 25 cm²

Enlarged Area = 15² = 225 cm² = 25 × 3²

Reduced Area = 2² = 4 cm² = 25 × 0.4²

(a) Enlarged: 15 cm side
(b) Reduced: 2 cm side
Final answer:

(a) Enlarged square has side length 15 cm
(b) Reduced square has side length 2 cm

Applied rules:

• Scale Factor > 1: Creates enlargement

• Scale Factor < 1: Creates reduction

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

Scaling Rules and Methods
\(\text{New Dimension} = \text{Original Dimension} \times \text{Scale Factor}\)
Scaling Formula
Scale Factor > 1
Enlargement
Figure becomes larger
Scale Factor = 1
No Change
Same size and shape
Scale Factor < 1
Reduction
Figure becomes smaller
Key definitions:

Scale Factor: The ratio of corresponding lengths in similar figures. It determines how much a figure is enlarged or reduced.

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

Enlargement: Scaling up a figure using a scale factor greater than 1.

Reduction: Scaling down a figure using a scale factor between 0 and 1.

Complete methodology:
  1. Identify Original Dimensions: Note all measurements of the original figure
  2. Determine Scale Factor: Identify whether it's an enlargement or reduction
  3. Apply Scaling: Multiply each dimension by the scale factor
  4. Calculate New Measurements: Find new area, perimeter, etc. if needed
  5. Verify Results: Check that all dimensions follow the same scale factor
Tip 1: When scale factor is less than 1, the new figure is smaller than the original.
Tip 2: Areas scale by the square of the scale factor.
Tip 3: Perimeters scale by the same factor as the lengths.
Tip 4: Angles remain unchanged in scaled figures.
Common errors: Forgetting to apply scale factor to all dimensions, confusing area and length scaling, mixing up enlargement and reduction.
Exam preparation: Practice with decimal and fractional scale factors, work with different geometric shapes, solve multi-step problems.
Formulas to know by heart:

• New Length = Original Length × Scale Factor

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

• New Perimeter = Original Perimeter × Scale Factor

• Scale Factor = New Measurement ÷ Original Measurement

• For enlargement: Scale Factor > 1

• For reduction: 0 < Scale Factor < 1

Solution: Exercises 4 to 5
4 Multi-Step Scaling Problem
Exercise 4
Rectangle ABCD has length 8 cm and width 5 cm. First, it is enlarged by a scale factor of 1.5. Then the resulting rectangle is reduced by a scale factor of 0.6. Find the final dimensions and compare to the original.
Definition:

Sequential Scaling: When multiple scale factors are applied in sequence, the overall effect is the product of all scale factors.

Original
8cm × 5cm
After Enlargement
12cm × 7.5cm
Final
7.2cm × 4.5cm
Step 1: Calculate after first scaling (enlargement)

New Length = 8 × 1.5 = 12 cm

New Width = 5 × 1.5 = 7.5 cm

Step 2: Calculate after second scaling (reduction)

Final Length = 12 × 0.6 = 7.2 cm

Final Width = 7.5 × 0.6 = 4.5 cm

Step 3: Find overall scale factor

Overall Scale Factor = 1.5 × 0.6 = 0.9

Step 4: Verify with overall scale factor

Final Length = 8 × 0.9 = 7.2 cm ✓

Final Width = 5 × 0.9 = 4.5 cm ✓

Step 5: Compare to original

Final dimensions are 90% of original dimensions

This is a slight reduction from the original

Final: 7.2cm × 4.5cm (overall SF = 0.9)
Final answer:

The final rectangle has dimensions 7.2 cm by 4.5 cm, which represents a 10% reduction from the original.

Applied rules:

• Sequential Scaling: Overall effect = product of individual scale factors

• Commutative Property: Order of operations doesn't affect final result

• Net Effect: Compare final result to original to understand total transformation

5 Area and Perimeter Scaling
Exercise 5
Pentagon ABCDE has an area of 40 cm² and a perimeter of 30 cm. It is scaled by a factor of 2.5. Find the area and perimeter of the new pentagon A'B'C'D'E'.
Definition:

Measurement Scaling: Different measurements scale differently: linear measurements by the scale factor, areas by the square of the scale factor.

Original
Area=40cm², Perim=30cm
Scale Factor
2.5
New
Area=250cm², Perim=75cm
Step 1: Calculate new perimeter

Perimeter scales linearly with the scale factor

New Perimeter = Original Perimeter × Scale Factor

New Perimeter = 30 × 2.5 = 75 cm

Step 2: Calculate new area

Area scales with the square of the scale factor

New Area = Original Area × (Scale Factor)²

New Area = 40 × 2.5² = 40 × 6.25 = 250 cm²

Step 3: Verify the relationships

Perimeter Ratio = 75/30 = 2.5 ✓

Area Ratio = 250/40 = 6.25 = 2.5² ✓

Step 4: Understand the difference

Linear measurements (sides, perimeter) scale by SF

2D measurements (areas) scale by SF²

New Area = 250cm², New Perimeter = 75cm
Final answer:

The new pentagon has an area of 250 cm² and a perimeter of 75 cm.

Applied rules:

• Linear Scaling: Lengths and perimeters scale by SF

• Area Scaling: Areas scale by SF²

• Dimensional Scaling: n-dimensional measurements scale by SF^n

Complete Guide: Scaling Figures Up and Down
\(\text{New Measurement} = \text{Original} \times \text{SF}^n\)
General Scaling Formula (n = dimension)
Key definitions:

Scale Factor (SF): The constant multiplier that determines how much a figure is enlarged or reduced. SF > 1 means enlargement, 0 < SF < 1 means reduction.

Enlargement: A transformation that creates a larger version of the original figure (SF > 1).

Reduction: A transformation that creates a smaller version of the original figure (0 < SF < 1).

Similarity: The property of figures having the same shape but different sizes.

Complete methodology:
  1. Identify the Original Figure: Note all dimensions, area, perimeter, etc.
  2. Determine the Scale Factor: Understand if it's an enlargement or reduction
  3. Apply to Linear Measurements: Multiply lengths by the scale factor
  4. Apply to Area Measurements: Multiply by (scale factor)²
  5. Apply to Volume Measurements: Multiply by (scale factor)³ (Grade 8+)
  6. Verify Consistency: Ensure all calculations follow the scaling rules
Tip 1: Remember that area scales by SF² and volume by SF³, not by SF.
Tip 2: When scale factor is a fraction, multiply by the reciprocal to go back to original.
Tip 3: Draw both the original and scaled figures to visualize the transformation.
Tip 4: Always check that all corresponding parts follow the same scale factor.
Common errors: Confusing area and length scaling, applying different scale factors to different dimensions, forgetting that angles remain unchanged.
Exam preparation: Practice with various scale factors including decimals and fractions, work with complex shapes, solve problems involving multiple transformations.
Formulas to know by heart:

• New Length = Original Length × Scale Factor

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

• New Perimeter = Original Perimeter × Scale Factor

• New Volume = Original Volume × (Scale Factor)³ (for 3D figures)

• Enlargement: Scale Factor > 1

• Reduction: 0 < Scale Factor < 1

• No Change: Scale Factor = 1

Visualizing Scaling Effects: Size Transformation Relationships
Exercise 6: Scaling Impact Analysis
Consider how different scale factors affect geometric measurements:
Square (side=4cm), Scale Factors: 0.5, 1, 1.5, 2, 2.5
Showing how length, area, and perimeter change

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

  • Linear dimensions scale directly with the scale factor
  • Areas scale with the square of the scale factor
  • Perimeters scale directly with the scale factor
  • Enlargements (SF > 1) increase size, reductions (SF < 1) decrease size

Questions & Answers

Question: Why does area scale by the square of the scale factor while length only scales by the scale factor?

Answer: This happens because area is a two-dimensional measurement while length is one-dimensional:

For length: Only one dimension is affected, so New Length = Original Length × Scale Factor

For area: Both length and width are affected, so:

  • New Area = (Original Length × SF) × (Original Width × SF)
  • New Area = Original Length × Original Width × SF²
  • New Area = Original Area × SF²

For example, if you have a 3×4 rectangle (area=12) and scale by factor 2:

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

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

Question: Can scale factors be negative? What would that mean?

Answer: While we typically work with positive scale factors in Grade 7, negative scale factors do exist in more advanced mathematics:

Positive Scale Factor: Maintains the orientation of the figure

Negative Scale Factor: Creates a figure that is also reflected or rotated

  • SF = -1: Creates a congruent figure rotated 180°
  • SF = -2: Creates an enlarged figure rotated 180°
  • SF = -0.5: Creates a reduced figure rotated 180°

However, in Grade 7, we focus on positive scale factors. Negative scale factors involve transformations beyond basic scaling and require knowledge of coordinate geometry and rotations.

Question: What happens to the angles when I scale a figure up or down?

Answer: The angles remain unchanged when you scale a figure! This is a fundamental property of similar figures:

Key Points:

  • Scaling changes the size but not the shape of a figure
  • All corresponding angles in similar figures are equal
  • Only the lengths of sides change according to the scale factor

For example, if you have a triangle with angles 30°, 60°, and 90°, any scaled version of this triangle will still have the same three angles, even though the side lengths will be different.

This preservation of angle measures is what ensures the scaled figure has the same shape as the original.