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.

Build · I Do / We Do / You Do

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.

Scale factor k = DE/AB = 9/6 = 1.5; multiply each matching side by 1.5.

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.

Stuck? Revisit lesson § "Scale Factor", always divide new by original, never the other way around.

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 ______.

Stuck? Revisit lesson § "Scale Factor", k > 1 means enlargement, 0 < k < 1 means reduction.

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

Stuck on 3.8? Volume ratio = 125/8. The linear scale factor is the cube root.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers, Do not peek before attempting

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.