Mathematics • Year 9 • Unit 3 • Lesson 1
Pythagorean Triples in the Real World
Apply $a^2 + b^2 = c^2$ to everyday right-angle situations: Egyptian rope triangles, kite frames, surveyors' triangles, ladders against walls, and TV screens. Then explain in your own words why the rule only works for right-angled triangles.
1. Word problems
Each problem uses $a^2 + b^2 = c^2$ — usually as a check that three lengths really do form a right-angled triangle. Show your working — a single final answer with no working only earns half marks.
1.1 — Egyptian rope triangle. Egyptian builders tied 12 evenly-spaced knots in a rope and folded it into a triangle with sides 3, 4 and 5 knot-lengths.
(a) Use Pythagoras to confirm the triangle is right-angled.
(b) Between which two sides does the right angle sit? 3 marks
1.2 — Kite frame. A diamond-shaped kite has its two cross-struts at right angles. The horizontal strut is 16 cm long; the vertical strut is 30 cm long. Each panel (quarter of the kite) is a right triangle with legs of 8 cm and 15 cm — half of each strut.
(a) Use Pythagoras to find the length of the slanted edge of one panel.
(b) Which common Pythagorean triple does each panel match? 3 marks
1.3 — Surveyors' check. A surveyor lays out a rectangular building site and measures the two adjacent sides as 9 m and 12 m. To check the corner is exactly square (90°), they measure the diagonal and get 15 m.
(a) Use Pythagoras to verify the corner really is right-angled.
(b) Identify which Pythagorean triple (and which multiple of it) is at work. 3 marks
1.4 — Ladder against a wall. A safety guide says a ladder is set up correctly when its foot is 7 m from the wall and its top reaches 24 m up the wall.
(a) Use Pythagoras to find the exact length of the ladder.
(b) Which triple does this match? 3 marks
1.5 — TV screen check. A salesperson advertises a "right-angled" rectangular TV. The width is 16 inches and the height is 12 inches, and they claim the diagonal is exactly 20 inches.
(a) Use Pythagoras to check whether the corners really are right angles (i.e. whether 12-16-20 satisfies the theorem).
(b) Which standard triple is 12-16-20 a multiple of? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A friend tries to use Pythagoras' theorem on a triangle with sides 4 cm, 5 cm and 6 cm. They compute $4^2 + 5^2 = 16 + 25 = 41$ and $6^2 = 36$, see that $41 \ne 36$, and conclude "Pythagoras' theorem is broken for this triangle."
In your own words, explain (i) what they have actually shown, (ii) which rule from Lesson 1 they have forgotten, and (iii) why the test still gives useful information about this triangle.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Egyptian rope triangle
(a) Check $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓. By the converse of Pythagoras' theorem, the triangle is $\mathbf{right\text{-}angled}$.
(b) The right angle is OPPOSITE the hypotenuse (length 5), so it sits $\mathbf{between\ the\ sides\ of\ length\ 3\ and\ 4}$.
This is why Egyptian builders used the 12-knot rope — folding it into a 3-4-5 triangle guarantees a perfect 90° corner.
1.2 — Kite frame
(a) $8^2 + 15^2 = 64 + 225 = 289$, so the slanted edge $= \sqrt{289} = \mathbf{17}$ cm.
(b) This is the $\mathbf{8\text{-}15\text{-}17}$ triple — one of the four to memorise.
1.3 — Surveyors' check
(a) $9^2 + 12^2 = 81 + 144 = 225 = 15^2$ ✓. The corner $\mathbf{is\ right\text{-}angled}$.
(b) Divide everything by 3: $9, 12, 15 = 3 \times (3, 4, 5)$ — so this is the $\mathbf{3\text{-}4\text{-}5}$ triple tripled.
Real-world note: this "3-4-5 method" is exactly how carpenters and builders still check square corners on building sites today.
1.4 — Ladder against a wall
(a) $7^2 + 24^2 = 49 + 576 = 625$, so the ladder length $= \sqrt{625} = \mathbf{25}$ m.
(b) This is the $\mathbf{7\text{-}24\text{-}25}$ triple — the fourth one to memorise.
(A 25 m ladder is unrealistically long — but the numbers make a clean triple, which is why this question shows up so often in textbooks.)
1.5 — TV screen check
(a) $12^2 + 16^2 = 144 + 256 = 400 = 20^2$ ✓. So yes, $\mathbf{the\ corners\ are\ right\ angles}$.
(b) Divide every side by 4: $12, 16, 20 = 4 \times (3, 4, 5)$ — so it's the $\mathbf{3\text{-}4\text{-}5}$ triple quadrupled.
Manufacturers love multiples of 3-4-5 because they give clean diagonal sizes.
2.1 — Explain your thinking (sample response)
(i) The friend has correctly shown that $4^2 + 5^2 \ne 6^2$. That's a true calculation. (ii) But they have forgotten that Pythagoras' theorem only applies to right-angled triangles — it isn't a rule about every triangle. Their triangle isn't broken; it simply isn't right-angled. (iii) The test still gives useful information: because $a^2 + b^2 \ne c^2$, we can conclude by the converse of Pythagoras that the 4-5-6 triangle has $\mathbf{no\ right\ angle}$. So the test acts like a yes/no detector for "is this triangle right-angled?"
Marking: 1 for naming the right-angle restriction; 1 for explaining the test still tells us something useful (it's a detector); 1 for the correct conclusion (no right angle); 1 for a clear, full-sentence explanation.