Mathematics • Year 8 • Unit 3 • Lesson 3
Pythagoras in the World
Bigger scenarios from everyday life: a bushwalking shortcut, a sportsfield diagonal, a treehouse rope, a delivery drone and a basketball court. Draw, label, and apply Pythagoras — then explain your thinking.
1. Word problems
Each problem hides a right triangle. Sketch, label, identify the right angle, and apply Pythagoras. Show full working — a single number with no diagram or working earns half marks at most.
1.1 — Bushwalking shortcut. A bushwalker leaves the campsite and walks 9 km due east along a fire trail, then turns and walks 12 km due north along a creek. She wants to head straight back to the campsite through the bush.
(a) Sketch the journey, marking the right angle.
(b) How far is the direct distance back to camp?
(c) How many kilometres does the shortcut save? 3 marks
1.2 — Basketball court diagonal. An FIBA basketball court is 28 m long and 15 m wide. The official line judge stands at one corner and watches a ball roll diagonally to the opposite corner.
(a) How far does the ball roll? (Round to 2 decimal places.)
(b) Is the diagonal more or less than 32 m? 3 marks
1.3 — Treehouse rope ladder. A treehouse platform is 4 m off the ground. A rope ladder hangs from the platform to a point on the ground 3 m away from the trunk.
(a) Sketch the treehouse, ground and rope as a right triangle.
(b) How long is the rope ladder?
(c) Identify the Pythagorean triple. 3 marks
1.4 — Delivery drone. A delivery drone flies from a warehouse at grid coordinates (2, 3) to a customer at (14, 8). Each grid unit is 1 km.
(a) Calculate Δx and Δy.
(b) Find the straight-line distance the drone flies, to 2 decimal places. 3 marks
1.5 — Skateboard ramp. A skateboarder builds a ramp. The vertical back-piece is 0.9 m tall and the ramp's slanted surface (the hypotenuse) is 1.5 m long.
(a) Sketch the ramp showing the slant, back-piece and ground.
(b) How far along the ground does the ramp extend? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked: "A truck driver drives 8 km north, then 6 km east, then 8 km north again. How far is the driver from the start?" The classmate writes: "She drove a total of 8 + 6 + 8 = 22 km, so she's 22 km from start." In your own words, explain (i) what mistake they have made, (ii) the correct method (find the total north and east displacements, then apply Pythagoras), and (iii) the correct final answer with working. Use the phrase "total distance is not the same as straight-line distance" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bushwalking shortcut
(a) Right triangle: east leg 9 km, north leg 12 km, hypotenuse = direct distance.
(b) c² = 9² + 12² = 81 + 144 = 225, so direct distance = 15 km.
(c) Original path = 9 + 12 = 21 km. Shortcut saves 21 − 15 = 6 km.
1.2 — Basketball court diagonal
(a) c² = 28² + 15² = 784 + 225 = 1009, so c = √1009 ≈ 31.76 m.
(b) Less than 32 m (31.76 < 32, since 32² = 1024 > 1009).
1.3 — Treehouse rope ladder
(a) Right triangle: vertical 4 m (treehouse), horizontal 3 m (ground), hypotenuse = rope length.
(b) c² = 4² + 3² = 16 + 9 = 25, so rope = 5 m long.
(c) 3-4-5 Pythagorean triple.
1.4 — Delivery drone
(a) Δx = 14 − 2 = 12 km; Δy = 8 − 3 = 5 km.
(b) d² = 12² + 5² = 144 + 25 = 169, so d = 13.00 km exactly (to 2 d.p.). (5-12-13 triple.)
1.5 — Skateboard ramp
(a) Right triangle: vertical back 0.9 m, ground (missing leg), slant = hypotenuse 1.5 m.
(b) a² = 1.5² − 0.9² = 2.25 − 0.81 = 1.44, so ramp extends 1.2 m along the ground. (0.9-1.2-1.5 = 3-4-5 × 0.3.)
2.1 — Explain your thinking (sample response)
The classmate has added all three travel distances together — but total distance is not the same as straight-line distance. The correct method is: first add up all the north travel (8 + 8 = 16 km) and all the east travel (6 km) separately, then use Pythagoras with the resulting right triangle. The displacement is d² = 16² + 6² = 256 + 36 = 292, so d = √292 ≈ 17.09 km. The driver travelled 22 km on the road but is only 17.09 km from her start in a straight line.
Marking: 1 mark for spotting the "added all distances" mistake; 1 mark for combining the displacements (north = 16, east = 6); 1 mark for applying Pythagoras to get ≈ 17.09 km; 1 mark for clear explanation using "total distance is not the same as straight-line distance".