Mathematics • Year 8 • Unit 3 • Lesson 1
Pythagoras — Mixed Challenge
Pull everything from Lesson 1 together: finding hypotenuses, recognising triples, scaled triples and decimal answers. Six mixed problems, one "find the mistake", and one open-ended Pythagorean-triple hunt.
1. Mixed problems — choose the right move
Each question pulls a different idea from Lesson 1. Decide which approach applies before you start writing. Show your working. 3 marks each
1.1 A right-angled triangle has legs a = 20 cm and b = 21 cm. Find the hypotenuse c.
1.2 A square has side length 5 cm. Find the length of its diagonal to 2 decimal places.
1.3 Is 9, 40, 41 a Pythagorean triple? Show your check using a² + b² = c².
1.4 A right-angled triangle has legs 0.5 m and 1.2 m. Find the hypotenuse. (Hint: identify the scaled triple first.)
1.5 A farm gate is 1.2 m wide and 0.9 m tall. Find the length of its diagonal brace, and identify the triple used.
1.6 A rectangle is 16 cm wide and 30 cm tall. Find the length of its diagonal (this is an 8-15-17 family — spot the multiplier).
2. Find the mistake
A Year 8 student has tried to find the hypotenuse of a right-angled triangle with legs 9 cm and 12 cm. 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
Student's working — find c when a = 9 and b = 12:
Line 1: c² = a² + b²
Line 2: c² = 9² + 12² = 81 + 144 = 225
Line 3: c = 225
Line 4: So the hypotenuse is 225 cm.
(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? Revisit lesson § Card 9 — "Stopping at c²" is one of the most common pitfalls. c² = 225 does NOT mean c = 225.3. Open-ended challenge — Pythagorean triple hunter
This question has more than one valid answer. 4 marks
3.1 A Pythagorean triple is a set of three whole numbers a, b, c (with a < b < c) such that a² + b² = c². The three "core" triples from the lesson are 3-4-5, 5-12-13 and 8-15-17.
Find three more Pythagorean triples where the hypotenuse is less than 30. They must NOT be a, b, c all equal to the core triples above (but multiples of the core triples ARE allowed, as long as the hypotenuse is < 30).
For each triple you find:
(i) Write it as a-b-c.
(ii) Show the check: a² + b² = c².
(iii) State whether it is a multiple of one of the core triples, OR a new "primitive" triple.
Bonus: at least one of your three triples must be a primitive (not a multiple of a core triple).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Legs 20 and 21
c² = 20² + 21² = 400 + 441 = 841, so c = √841 = 29 cm. (20-21-29 is a primitive Pythagorean triple.)
1.2 — Square with side 5
Diagonal² = 5² + 5² = 25 + 25 = 50, so diagonal = √50 ≈ 7.07 cm (to 2 d.p.).
1.3 — Is 9, 40, 41 a triple?
Check: 9² + 40² = 81 + 1600 = 1681 = 41². Yes — 9-40-41 IS a Pythagorean triple.
1.4 — Legs 0.5 and 1.2
0.5-1.2-? is 5-12-13 × 0.1. So c = 13 × 0.1 = 1.3 m. Check: 0.25 + 1.44 = 1.69 = 1.3² ✓
1.5 — Farm gate
c² = 1.2² + 0.9² = 1.44 + 0.81 = 2.25, so diagonal = √2.25 = 1.5 m. This is the 3-4-5 triple scaled by 0.3 (0.9-1.2-1.5).
1.6 — Rectangle 16 × 30
Triple identification: 16 = 8 × 2 and 30 = 15 × 2, so the family is 8-15-17 × 2 = 16-30-34. Diagonal = 34 cm. Check: 256 + 900 = 1156 = 34² ✓
2 — Find the mistake
(a) The mistake is on Line 3 (carried into Line 4).
(b) The student wrote "c = 225" but Line 2 only gave c² = 225, not c itself. They forgot the final step — taking the square root. c² = 225 means c = √225, not 225.
(c) Corrected working:
c² = a² + b² = 9² + 12² = 81 + 144 = 225
c = √225 = 15 cm. ✓
Sanity check: 9-12-15 is a ×3 multiple of the 3-4-5 triple. The hypotenuse is 15, not 225 (which would be 15 squared).
3 — Pythagorean triple hunter (sample solution)
There are many valid sets. Three good examples (all with c < 30):
Triple 1: 6-8-10.
Check: 36 + 64 = 100 = 10² ✓. This is the 3-4-5 triple × 2 — a multiple.
Triple 2: 9-12-15.
Check: 81 + 144 = 225 = 15² ✓. This is the 3-4-5 triple × 3 — a multiple.
Triple 3: 7-24-25.
Check: 49 + 576 = 625 = 25² ✓. This is a primitive (not a multiple of any core triple).
Other valid answers: 12-16-20 (3-4-5 × 4), 10-24-26 (5-12-13 × 2), 15-20-25 (3-4-5 × 5), 16-30-? (No — 34 > 30, so excluded), 20-21-29 (a primitive!), 9-40-? (No — 41 > 30, excluded).
Marking: 1 mark per valid distinct triple with correct check (up to 3 marks). 1 bonus mark for including at least one primitive (e.g. 7-24-25 or 20-21-29), not just multiples of 3-4-5.