Mathematics • Year 10 • Unit 3 • Lesson 11
Similarity and Scale Factors — Skill Drill
Build fluency with the core idea from Lesson 11: lengths scale by k, areas scale by k², volumes scale by k³. Calculate scale factors as ratios (never differences), and work both directions — from given lengths to k, and from k back to unknown lengths, areas or volumes.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Triangle ABC is similar to triangle DEF. AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 9 cm. Find EF and DF.
Step 1 — Find the scale factor.
k = DE / AB = 9 / 6 = 1.5
Reason: scale factor = new ÷ original. DE is the new side that corresponds to AB.
Step 2 — Apply k to each remaining side.
EF = BC × k = 8 × 1.5 = 12 cm
DF = AC × k = 10 × 1.5 = 15 cm
Reason: every length in the image triangle is 1.5 times the corresponding length in the original.
Step 3 — Verify by checking all three ratios.
9/6 = 12/8 = 15/10 = 1.5 ✓
Reason: all three ratios must equal the same k. If they do not, you have made an error.
Answer: EF = 12 cm, DF = 15 cm.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. Triangle PQR is similar to triangle XYZ. PQ = 4 cm, QR = 5 cm, PR = 7 cm, and XY = 12 cm. Find YZ and XZ.
Step 1 — Find the scale factor:
k = XY / PQ = ______ / ______ = ______
Step 2 — Use k to find YZ:
YZ = QR × k = ______ × ______ = ______ cm
Step 3 — Use k to find XZ:
XZ = PR × k = ______ × ______ = ______ cm
Step 4 — Verify all three ratios are equal:
XY/PQ = ______ , YZ/QR = ______ , XZ/PR = ______
Step 5 — Enlargement or reduction? k is ______ 1, so this is an ______.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — calculate scale factors
3.1 Two corresponding sides of similar figures are 4 cm and 12 cm. Find the scale factor k (image ÷ original). 1 mark
3.2 Two corresponding sides are 9 cm (original) and 3 cm (image). Find k, then state whether this is an enlargement or a reduction. 1 mark
3.3 Triangle ABC ~ triangle DEF with scale factor k = 3. BC = 6 cm. Find EF. 1 mark
3.4 If the linear scale factor is k = 4, write down the area scale factor and the volume scale factor. 1 mark
Standard — area and volume scaling
3.5 A rectangle is enlarged so every length triples (k = 3). The original area is 50 cm². Find the new area, showing the k² step. 2 marks
3.6 Two similar cubes have side lengths 2 cm and 6 cm. By what factor does the volume increase? Show k, then k³. 2 marks
Extension — work backwards
3.7 Two similar triangles have areas 36 cm² and 144 cm². The shortest side of the smaller triangle is 4 cm. Find the shortest side of the larger triangle. (Hint: k = √(area ratio).) 3 marks
3.8 Two similar spheres have volumes 8 cm³ and 125 cm³. Find the linear scale factor k. (Hint: k = ∛(volume ratio).) 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (PQR → XYZ)
Step 1: k = 12 / 4 = 3.
Step 2: YZ = 5 × 3 = 15 cm.
Step 3: XZ = 7 × 3 = 21 cm.
Step 4: 12/4 = 3, 15/5 = 3, 21/7 = 3 — all equal ✓.
Step 5: k is greater than 1, so this is an enlargement.
3.1 — Scale factor 4 → 12
k = 12 / 4 = 3.
3.2 — 9 cm (original) → 3 cm (image)
k = 3 / 9 = 1/3. Since 0 < k < 1, this is a reduction.
3.3 — BC = 6, k = 3
EF = BC × k = 6 × 3 = 18 cm.
3.4 — k = 4
Area scale factor = k² = 16. Volume scale factor = k³ = 64.
3.5 — Rectangle area, k = 3
Area scale factor = k² = 3² = 9.
New area = 50 × 9 = 450 cm².
Length triples but area gets nine times bigger — this is the most-tested idea in the lesson.
3.6 — Cubes 2 cm and 6 cm
k = 6 / 2 = 3.
Volume scale factor = k³ = 3³ = 27.
Check: 2³ = 8, 6³ = 216, and 216 / 8 = 27 ✓.
3.7 — Triangles with areas 36 and 144
Area ratio = 144 / 36 = 4.
Linear scale factor k = √4 = 2.
Shortest side of larger triangle = 4 × 2 = 8 cm.
3.8 — Spheres with volumes 8 and 125
Volume ratio = 125 / 8.
k = ∛(125/8) = ∛125 / ∛8 = 5 / 2 = 5/2 (i.e. 2.5).
The cube root undoes the cubing — same logic as square root for areas.