Mathematics • Year 7 • Unit 3 • Lesson 16
Similar Figures — Mixed Challenge
Combine everything: scale factor, vertex order, similarity vs congruence, and the difference between equal ratios and equal differences. Spot a plausible mistake and tackle an open-ended challenge.
1. Mixed problems
Each question mixes ideas from this lesson. Show your working. 2 marks each
1.1 Two similar triangles. Small has sides 6, 8, 10. Large has corresponding sides 9, 12, 15. Find the scale factor.
1.2 △PQR ∼ △LMN. ∠P = 40°, ∠Q = 75°. Find ∠N.
1.3 Two squares — one with side 5 cm, one with side 12 cm. Are they similar? Justify.
1.4 A triangle has sides 5, 7, 9. A similar enlarged triangle has shortest side 20. Find the scale factor and the other two sides.
1.5 Two figures are similar with scale factor 1. What special name do we give to such a pair? Why?
1.6 A rectangle has sides 4 cm × 9 cm. List THREE different rectangles that are similar to it (give each one's two side lengths and the scale factor from the original).
2. Find the mistake
Another Year 7 student tried to decide whether two triangles are similar. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: Triangle A has sides 4, 6, 8. Triangle B has corresponding sides 6, 9, 10. Are they similar?
Line 1: Pair the sides shortest to shortest: 4 ↔ 6, 6 ↔ 9, 8 ↔ 10.
Line 2: First ratio: 6 ÷ 4 = 1.5.
Line 3: Second ratio: 9 ÷ 6 = 1.5. Both match so far!
Line 4: Since two ratios match, the triangles are similar. SF = 1.5.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Compute the THIRD ratio (10 ÷ 8). Does it equal 1.5? If not, the figures aren't similar — even if two ratios matched.3. Open-ended challenge — design a family of similar triangles
This question has many correct answers. Show your work clearly. 4 marks
3.1 Start with a right-angled triangle with sides 3, 4, 5 (a "3-4-5 triangle"). Design three different triangles that are similar to it, using these three scale factors: SF = 2, SF = 4, and SF = 0.5 (a reduction).
For each new triangle:
(i) List the three new side lengths.
(ii) Show one ratio check (new ÷ old) to confirm the scale factor.
(iii) State whether the new triangle is an enlargement or a reduction.
Bonus: Find a fourth similar triangle whose three sides are all even whole numbers (other than the SF = 2 one). State the scale factor.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Two similar triangles
Check ratios: 9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5, 15 ÷ 10 = 1.5. All match. SF = 1.5.
1.2 — Angle from similarity statement
△PQR ∼ △LMN means P ↔ L, Q ↔ M, R ↔ N. Similar triangles have equal corresponding angles. So ∠N = ∠R. In △PQR: ∠R = 180 − 40 − 75 = 65°. So ∠N = 65°.
1.3 — Two squares
All squares have four right angles and four equal sides. Corresponding sides are 5 ↔ 12. Both pairs of sides give the same ratio 12 ÷ 5 = 2.4 (and the angles all equal 90°). Yes — similar with SF = 2.4. In fact, any two squares are similar.
1.4 — Scale up to shortest side 20
SF = 20 ÷ 5 = 4. Other two sides: 7 × 4 = 28 and 9 × 4 = 36.
1.5 — SF = 1
The figures are congruent (symbol ≡) — same shape AND same size. Congruent is a special case of similar where the scale factor is exactly 1.
1.6 — Three similar rectangles to 4 × 9
Many answers. Examples (your three may differ — any consistent SF works):
SF = 2: 8 × 18.
SF = 0.5: 2 × 4.5.
SF = 3: 12 × 27.
Check the ratios are 4 : 9 (= 8 : 18 = 2 : 4.5 = 12 : 27). All ✓.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) For similarity, ALL THREE pairs of corresponding sides must have the same ratio. The student stopped after only TWO ratios — they never checked the third pair (10 ÷ 8).
(c) Corrected working:
Line 1: Pair sides: 4 ↔ 6, 6 ↔ 9, 8 ↔ 10. (unchanged)
Line 2: 6 ÷ 4 = 1.5. (unchanged)
Line 3: 9 ÷ 6 = 1.5. (unchanged)
Line 4 (fixed): Check the third ratio: 10 ÷ 8 = 1.25.
Line 5 (new): 1.5 ≠ 1.25, so the ratios are NOT all equal.
Line 6 (new): The triangles are NOT similar.
The fix: never stop early — check EVERY pair.
3 — Open-ended challenge (sample solutions)
Original triangle sides: 3, 4, 5.
SF = 2 (enlargement): sides = 6, 8, 10. Check: 6 ÷ 3 = 2 ✓. Enlargement.
SF = 4 (enlargement): sides = 12, 16, 20. Check: 12 ÷ 3 = 4 ✓. Enlargement.
SF = 0.5 (reduction): sides = 1.5, 2, 2.5. Check: 1.5 ÷ 3 = 0.5 ✓. Reduction.
Bonus — fourth similar triangle with all-even sides: Try SF = 6 — sides become 18, 24, 30 (all even). Or SF = 8 — sides become 24, 32, 40. Many answers; any even SF works because 3 × (even) = even.
Marking: 1 mark per correct triangle (3 marks total), 1 mark for any valid bonus answer with the SF stated.