Mathematics • Year 8 • Unit 3 • Lesson 1
Pythagoras in the Real World
Use c² = a² + b² where it actually shows up: TV screens, builders' corners, ladders, soccer fields and bike rides. Then explain your thinking in your own words.
1. Word problems
Each problem hides a right-angled triangle. Draw a quick sketch, label sides, then apply c² = a² + b². Show working — a single final answer with no working earns only half marks.
1.1 — The builder's corner trick. A builder marks 3 m along one wall and 4 m along the adjoining wall. To check that the corner is exactly 90°, what should the diagonal between the two marks measure?
(a) Sketch the right-angled triangle.
(b) Use Pythagoras to calculate the diagonal.
(c) Why does this trick guarantee a right angle? 3 marks
1.2 — TV screen diagonal. Modern TVs are advertised by the length of their diagonal (e.g. "55-inch TV"). A new flat-screen is 48 cm wide and 27 cm tall.
(a) Find the screen's diagonal length to 1 decimal place.
(b) Roughly how many inches is that? (1 inch ≈ 2.54 cm.) 3 marks
1.3 — Park shortcut. A rectangular park is 60 m long and 80 m wide. Aiden walks from one corner straight across to the opposite corner. His friend Maria walks around two sides of the park instead.
(a) How far does Aiden walk (the diagonal)?
(b) How far does Maria walk?
(c) How much shorter is Aiden's path? 3 marks
1.4 — Bike ride. Lucas rides 12 km east, then turns 90° and rides 5 km north.
(a) Sketch the path and label all distances.
(b) What is the straight-line distance from his start to his end point?
(c) Identify the Pythagorean triple. 3 marks
1.5 — Tile the floor. A square tile has side length 30 cm. A worker lays the tile diagonally to check it fits a corner.
(a) Find the length of the tile's diagonal to 1 decimal place.
(b) Explain why the diagonal of any square is always longer than its sides. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate has been asked to find the hypotenuse of a triangle with legs 6 cm and 8 cm. They write "c = 6 + 8 = 14 cm". In your own words, explain (i) what mistake they have made, (ii) what the correct answer is and how to get it, and (iii) one quick "sanity check" that would have warned them their answer was wrong. Use the phrase "square first, then add" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Builder's corner trick
(a) Right triangle with legs 3 m and 4 m, diagonal = hypotenuse.
(b) c² = 3² + 4² = 9 + 16 = 25, so diagonal = 5 m.
(c) Pythagoras only gives c = 5 when the angle between the two legs is exactly 90°. If the diagonal measures 5 m, the corner MUST be a right angle.
1.2 — TV diagonal
(a) c² = 48² + 27² = 2304 + 729 = 3033, so c = √3033 ≈ 55.1 cm.
(b) 55.1 ÷ 2.54 ≈ 21.7 inches — close to a 22-inch TV.
1.3 — Park shortcut
(a) c² = 60² + 80² = 3600 + 6400 = 10 000, so Aiden's diagonal = 100 m.
(b) Maria walks 60 + 80 = 140 m.
(c) Aiden saves 140 − 100 = 40 m.
1.4 — Bike ride
(a) Right triangle: east leg 12 km, north leg 5 km, hypotenuse = straight-line distance.
(b) c² = 12² + 5² = 144 + 25 = 169, so c = √169 = 13 km.
(c) 5-12-13 is a core Pythagorean triple.
1.5 — Tile diagonal
(a) c² = 30² + 30² = 900 + 900 = 1800, so c = √1800 ≈ 42.4 cm (to 1 d.p.).
(b) The diagonal is the hypotenuse of a right triangle with both legs equal to the side length. Since the hypotenuse is always the longest side, the diagonal must be longer than either side. (Exactly side × √2 ≈ 1.414 times the side.)
2.1 — Explain your thinking (sample response)
The classmate has added the two leg lengths instead of squaring them first. Pythagoras' Theorem says c² = a² + b², so we must square first, then add: c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm. A quick sanity check: the hypotenuse is the longest side of a right triangle, but it cannot be longer than the two legs added together (otherwise the triangle wouldn't close). So if 6 + 8 = 14, the real answer must be less than 14 — and 10 fits, but 14 itself definitely cannot be the hypotenuse.
Marking: 1 mark for spotting the "added instead of squared" mistake; 1 mark for showing 6² + 8² = 100 and c = 10; 1 mark for the sanity check (or any valid check); 1 mark for clear full-sentence explanation using "square first, then add".