Advanced Scale factors

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Part 1: Apply and reverse scale factors

1. A length of 4 cm is enlarged by a scale factor of 2.5. What is the corresponding length?

6.5 cm

8 cm

10 cm

12.5 cm

2. Which scale factor represents a reduction?

0.25

1.5

3. An image has a corresponding length of 18 cm after an enlargement by a scale factor of 3. Find the original length.

4. A drawing has corresponding lengths of 3 cm and 8 cm. A proposed image has lengths of 7.5 cm and 20 cm. Is the proposed image valid? Justify your answer using scale factors.

Part 2: Multi-step reasoning and checking

5. A shape has corresponding lengths of 6 cm, 10 cm and 14 cm. It is enlarged by a scale factor of 1.5. Find all three image lengths.

6. An image has corresponding lengths of 4.8 cm, 7.2 cm and 12 cm. It was made by reducing the original shape. The scale factor was 0.6. Find the three original lengths.

7. A shape is enlarged by ×3 and then reduced by ×1/3. What happens to each corresponding length? Explain why.

8. A length changes from 7.5 cm to 12.5 cm. Find the scale factor. Then use it to find the image length of a corresponding 9 cm side.

9. Check each proposed image. State whether it is valid and give a reason.

a) Original lengths: 5 cm and 8 cm. Image lengths: 15 cm and 24 cm. The corresponding angle is unchanged.

b) Original lengths: 6 cm and 10 cm. Image lengths: 15 cm and 26 cm. The corresponding angle is unchanged.

Part 3: Draw, measure and justify

10. Draw a triangle with side lengths 4 cm, 6 cm and 8 cm. Draw an enlarged image using a scale factor of 2.5, so its corresponding side lengths are 10 cm, 15 cm and 20 cm. Mark corresponding angles as unchanged. Check your drawing by measuring the sides and explaining how the ratios confirm the scale factor.

Answer key

1. 10 cm, because 4 × 2.5 = 10.

2. 0.25, because it is between 0 and 1.

3. 6 cm, because 18 ÷ 3 = 6.

4. Valid. Both corresponding lengths use a scale factor of 2.5: 7.5 ÷ 3 = 2.5 and 20 ÷ 8 = 2.5.

5. 9 cm, 15 cm and 21 cm. Each length is multiplied by 1.5.

6. 8 cm, 12 cm and 20 cm. Divide each image length by 0.6.

7. Each length returns to its original size. The combined scale factor is 3 × 1/3 = 1.

8. Scale factor = 12.5 ÷ 7.5 = 5/3. The 9 cm side becomes 15 cm, because 9 × 5/3 = 15.

9. a) Valid: 15 ÷ 5 = 3 and 24 ÷ 8 = 3, with angles unchanged. b) Invalid: 15 ÷ 6 = 2.5 but 26 ÷ 10 = 2.6, so the corresponding lengths do not share one scale factor.

10. The image should have side lengths 10 cm, 15 cm and 20 cm. Each image length divided by its original length should equal 2.5, and corresponding angles should be unchanged.