Mathematics • Year 7 • Unit 3 • Lesson 16
Introduction to Similar Figures
Build fluency with similar figures: same shape, possibly different size. Corresponding angles are equal and corresponding sides are in the same ratio — that ratio is the scale factor (SF = new ÷ old).
1. I do — fully worked example
Read every line. Each step shows why, not just what.
Problem. A small triangle has sides 3, 4, 5 cm. A larger triangle has corresponding sides 9, 12, 15 cm. Are they similar? If so, find the scale factor.
Step 1 — Match corresponding sides (shortest ↔ shortest, longest ↔ longest).
3 ↔ 9 4 ↔ 12 5 ↔ 15
Reason: in similar triangles the order of side lengths is preserved.
Step 2 — Compute each ratio (new ÷ old).
9 ÷ 3 = 3 12 ÷ 4 = 3 15 ÷ 5 = 3
Reason: scale factor must be the same for every pair of corresponding sides.
Step 3 — Compare the ratios.
All three ratios equal 3, so they match.
Reason: equal ratios across every pair ⇒ figures are similar.
Step 4 — State the answer.
The triangles ARE similar. SF = 3.
Reason: every side of the new triangle is 3 times the corresponding side of the old.
Answer: Similar; scale factor = 3.
2. We do — fill in the missing steps
Same structure as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Triangle ABC has sides 5, 7, 9 cm. Triangle DEF has corresponding sides 15, 21, 27 cm. Find the scale factor from ABC to DEF.
Step 1 — Match corresponding sides (shortest ↔ shortest, etc.):
5 ↔ _______ 7 ↔ _______ 9 ↔ _______
Step 2 — Compute each ratio (new ÷ old):
15 ÷ 5 = _______
21 ÷ 7 = _______
27 ÷ 9 = _______
Step 3 — Do all three ratios match? _______
Step 4 — State the answer:
Triangle ABC is similar to triangle DEF with SF = _______ .
3. You do — independent practice
Show your working under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Two similar rectangles. The smaller has shortest side 4 cm; the larger has shortest side 16 cm. Find the scale factor. 1 mark
3.2 △ABC ∼ △XYZ. Name the side of △XYZ that corresponds to side AC. 1 mark
3.3 A photo is 8 cm wide. It is enlarged to be 32 cm wide. What is the scale factor? 1 mark
3.4 Two figures are congruent. What is their scale factor? 1 mark
Standard — check the ratios
3.5 Triangle P has sides 4, 6, 8. Triangle Q has corresponding sides 6, 9, 12. Are they similar? If yes, find the scale factor. Show every ratio. 2 marks
3.6 Rectangle A is 5 cm × 8 cm. Rectangle B is 15 cm × 20 cm. Are they similar? Justify by checking ratios. 2 marks
Extension — push your thinking
3.7 In △KLM, ∠K = 50°, ∠L = 60°, ∠M = 70°. In △XYZ, ∠X = 70°, ∠Y = 50°, ∠Z = 60°. Write the similarity statement with the correct vertex order (e.g. △KLM ∼ △???). 2 marks
3.8 Triangle JKL has sides 6, 8, 11. Triangle MNP has corresponding sides 9, 12, 16. Without computing every ratio, decide whether they are similar by checking ONLY ONE pair of ratios — then explain why one pair is NOT enough to confirm similarity in general. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (ABC ↔ DEF, sides 5,7,9 and 15,21,27)
Step 1: 5 ↔ 15, 7 ↔ 21, 9 ↔ 27.
Step 2: 15 ÷ 5 = 3, 21 ÷ 7 = 3, 27 ÷ 9 = 3.
Step 3: Yes, all three ratios match.
Step 4: △ABC ∼ △DEF with SF = 3.
3.1 — Scale factor
SF = new ÷ old = 16 ÷ 4 = 4.
3.2 — Corresponding side to AC
△ABC ∼ △XYZ means A ↔ X, B ↔ Y, C ↔ Z. So AC corresponds to XZ.
3.3 — Enlarged photo
SF = 32 ÷ 8 = 4.
3.4 — Congruent figures
Congruent means identical (same shape AND size). SF = 1.
3.5 — Triangle P and Q
Ratios: 6 ÷ 4 = 1.5, 9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5. All three match.
Yes, similar. SF = 1.5 (or 3/2).
3.6 — Rectangles A and B
Match shortest ↔ shortest: 5 ↔ 15 and 8 ↔ 20.
Ratios: 15 ÷ 5 = 3 and 20 ÷ 8 = 2.5.
3 ≠ 2.5, so the rectangles are NOT similar. The two ratios must match for similarity.
3.7 — Similarity statement
Match the equal angles: ∠K = 50° = ∠Y, so K ↔ Y. ∠L = 60° = ∠Z, so L ↔ Z. ∠M = 70° = ∠X, so M ↔ X.
Vertex order must match: △KLM ∼ △YZX.
3.8 — Triangle JKL vs MNP
Check ratios: 9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5, 16 ÷ 11 ≈ 1.454…
The first two match but the third does NOT. NOT similar.
Why one pair isn't enough in general: two triangles can have the same ratio on one pair of sides but different ratios on others. ALL pairs of corresponding sides must give the SAME scale factor.