Mathematics • Year 10 • Unit 3 • Lesson 11
Similarity and Scale Factors — Mixed Challenge
Pull every idea from Lesson 11 together: calculate scale factors from lengths, from areas (k = √ratio), and from volumes (k = ∛ratio); apply k, k² and k³ in the correct direction; spot the classic ratio-vs-difference mistake; and design your own similar figures.
1. Mixed problems — choose the right idea
Each question pulls on a different idea from Lesson 11. Decide first whether you need k, k² or k³. 2-3 marks each
1.1 Triangle ABC ~ triangle DEF. AB = 5 cm, DE = 15 cm, BC = 7 cm, AC = 9 cm. Find EF and DF. 3 marks
1.2 Two similar prisms have surface areas 50 cm² and 200 cm². Find the linear scale factor k and the volume scale factor. 2 marks
1.3 A rectangular paddock is reduced on a plan with linear scale factor k = 1/200. The real paddock has area 5000 m². Find the area on the plan, in cm². 3 marks
1.4 Two similar cones have volumes 27 cm³ and 1000 cm³. The smaller cone has a slant height of 6 cm. Find the slant height of the larger cone. 3 marks
1.5 Two similar triangles have areas in the ratio 9:25. The longest side of the smaller triangle is 12 cm. Find the longest side of the larger triangle. 3 marks
1.6 A child's toy block is a cube of side 4 cm. A similar cube of side 6 cm is made for an older child. The smaller block weighs 64 g. Assuming the same material, find the weight of the larger block. 3 marks
2. Find the mistake
Another Year 10 student has tried to work out the new area after an enlargement. Their working is shown. Exactly one line contains a mistake. Spot it, explain why it is wrong, then re-do the working correctly. 3 marks
Problem the student tried: A rectangle has area 30 cm². It is enlarged with linear scale factor k = 4. Find the new area.
Line 1: Linear scale factor k = 4.
Line 2: Area scale factor = k² = 4² = 16.
Line 3: New area = 30 + 16 = 46 cm².
Line 4: Final answer: 46 cm².
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected line and give the correct new area.
Stuck? The area scale factor is a multiplier, not something you add to the original area.3. Open-ended challenge — design your own similar pair
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design two different rectangular prisms (boxes) such that:
- both prisms have integer side lengths,
- the prisms are similar to each other,
- the volume of the larger prism is exactly 27 times the volume of the smaller one.
For each prism, state the dimensions (length × width × height) and the volume. Then state the linear scale factor k that connects them, and verify k³ = 27.
Bonus: Explain in one sentence why the volume ratio of 27 forces k to be exactly 3.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Triangle ABC ~ DEF
k = DE / AB = 15 / 5 = 3.
EF = BC × k = 7 × 3 = 21 cm.
DF = AC × k = 9 × 3 = 27 cm.
1.2 — Surface areas 50 and 200
Area ratio = 200 / 50 = 4.
k = √4 = 2.
Volume scale factor = k³ = 8.
1.3 — Paddock plan, k = 1/200
Area scale factor = k² = (1/200)² = 1/40000.
Plan area = 5000 m² × (1/40000) = 5000/40000 m² = 0.125 m² = 1250 cm².
Note: 1 m² = 10000 cm², so 0.125 m² = 1250 cm².
1.4 — Cones with volumes 27 and 1000
Volume ratio = 1000 / 27.
k = ∛(1000/27) = 10/3.
Larger slant height = 6 × (10/3) = 20 cm.
1.5 — Triangle area ratio 9:25
k = √(25/9) = 5/3.
Larger longest side = 12 × (5/3) = 20 cm.
1.6 — Cubes of side 4 cm and 6 cm
k = 6 / 4 = 3/2.
Weight scale factor = k³ = (3/2)³ = 27/8.
Larger weight = 64 × (27/8) = 8 × 27 = 216 g.
Check: a 6 cm cube has volume 216 cm³; a 4 cm cube has volume 64 cm³; 216/64 = 27/8 ✓.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student has added the area scale factor instead of multiplying. Scale factors always multiply the original quantity — they are never added. Lesson 11's "Common Pitfalls" warns about this exact error.
(c) Corrected: New area = 30 × 16 = 480 cm².
3 — Open-ended (sample solutions)
Volume ratio = 27 means k³ = 27, so k = ∛27 = 3. So both prisms must be similar with linear scale factor 3.
Prism A (smaller): 2 cm × 3 cm × 5 cm. Volume = 30 cm³.
Prism B (larger): 6 cm × 9 cm × 15 cm (every side ×3). Volume = 810 cm³.
Check: 810 / 30 = 27 ✓. Linear scale factor k = 3, so k³ = 27 ✓.
Bonus: Because volume scales by k³, asking for a volume ratio of 27 is the same as asking for k³ = 27, and the only positive real number with cube equal to 27 is 3.
Marking: 1 for correctly stating k = 3, 1 for a valid smaller prism with integer sides, 1 for the larger prism dimensions correct, 1 for showing 27 = k³ and the bonus reasoning. Any pair of integer-sided prisms scaled by 3 is acceptable.