Mathematics • Year 8 • Unit 3 • Lesson 3
Pythagoras Applications — Mixed Challenge
Mix navigation, rectangle diagonals, grid distances, roof rafters and multi-step problems. Six mixed questions, one "find the mistake", and one open-ended design challenge.
1. Mixed problems — different setups, same theorem
Each question is a different application. Always draw a quick sketch before calculating. Show working. 3 marks each
1.1 A plane flies 80 km east and 60 km north. Find the straight-line distance from its starting point.
1.2 A rectangular soccer pitch is 100 m long and 64 m wide. Find the diagonal to 1 decimal place.
1.3 Find the distance between points P(−2, 1) and Q(4, 9) on a grid where each unit is 1 metre.
1.4 A 12 m roof has its ridge 2.5 m above the centre. Find the length of one rafter (slant side) to 2 decimal places. (Hint: half-width = 6 m.)
1.5 A ship sails 6 km east, then turns 90° and sails 8 km north, then turns 90° and sails 6 km west. How far in a straight line is it from its starting point now? (Hint: the east and west cancel, leaving only the north displacement.)
1.6 A rectangular swimming pool is 10 m × 24 m. A diagonal rope is stretched across it. Pool tiles cost $45 per metre of edging. Calculate (a) the diagonal length, and (b) the cost of edging just the diagonal rope (no other sides).
2. Find the mistake
A student is asked to find the rafter length for a roof that is 14 m wide with a ridge 3 m above the centre. Their working is below. Exactly one line contains a mistake — spot it, explain why, and re-do the working. 3 marks
Student's working — find the rafter length:
Line 1: The right triangle has legs = 14 m (full roof width) and 3 m (ridge height).
Line 2: c² = 14² + 3² = 196 + 9 = 205
Line 3: c = √205 ≈ 14.32
Line 4: So each rafter is ≈ 14.32 m long.
(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 — "using the full width instead of the half-width for roof problems" is a common pitfall. The rafter goes from the ridge to ONE eave, not all the way across the roof.3. Open-ended challenge — design a delivery zone
This question has more than one valid answer. 4 marks
3.1 A pizza shop is located at the grid point (0, 0). Their delivery drone can fly anywhere within a straight-line distance of 10 km from the shop (the "delivery zone").
Find FOUR delivery addresses on a grid where:
- Each address has integer coordinates (x, y).
- Each address is exactly different distances from the shop (no two addresses can be the same distance away).
- All four addresses must lie inside the delivery zone (distance ≤ 10 km).
- At least one address must use a Pythagorean triple (giving an integer distance).
For each address, state the coordinates and the distance from the shop (show working with Pythagoras).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Plane 80 E, 60 N
c² = 80² + 60² = 6400 + 3600 = 10 000, so distance = 100 km. (60-80-100 = 3-4-5 × 20.)
1.2 — Soccer pitch 100 × 64
c² = 100² + 64² = 10 000 + 4096 = 14 096, so c = √14 096 ≈ 118.7 m (to 1 d.p.).
1.3 — P(−2, 1) to Q(4, 9)
Δx = 4 − (−2) = 6; Δy = 9 − 1 = 8. d² = 6² + 8² = 36 + 64 = 100, so d = 10 m. (6-8-10 = 3-4-5 × 2.)
1.4 — Rafter for 12 m roof
Half-width = 6 m, ridge height = 2.5 m. c² = 6² + 2.5² = 36 + 6.25 = 42.25, so rafter = √42.25 = 6.50 m (to 2 d.p.).
1.5 — Ship 6 E, 8 N, 6 W
Net east = 6 − 6 = 0; net north = 8. So the ship ends up directly 8 km north of its start: distance = 8 km. (Pythagoras gives c² = 0 + 64, c = 8.)
1.6 — Pool 10 × 24
(a) c² = 10² + 24² = 100 + 576 = 676, so diagonal = √676 = 26 m. (10-24-26 = 5-12-13 × 2.)
(b) Cost = 26 × $45 = $1170.
2 — Find the mistake
(a) The mistake is on Line 1 (carried through Lines 2–4).
(b) The student used the FULL roof width (14 m) as a leg, but the rafter only spans from the ridge to ONE side of the roof — so the horizontal leg should be the HALF-width: 14 ÷ 2 = 7 m. The ridge sits at the centre of the roof.
(c) Corrected working:
Right triangle: half-width = 7 m, ridge height = 3 m, rafter = hypotenuse.
c² = 7² + 3² = 49 + 9 = 58
c = √58 ≈ 7.62 m. ✓
Sanity check: each rafter (7.62 m) is much shorter than the full roof width (14 m) — correct geometry.
3 — Pizza delivery zone (sample solution)
Many valid sets. One good example (all distances ≤ 10 km, all different):
Address 1: (3, 4). d = √(9 + 16) = √25 = 5 km. ✓ Uses 3-4-5 triple (integer distance).
Address 2: (6, 8). d = √(36 + 64) = √100 = 10 km. ✓ Right at the edge of the zone; 6-8-10 triple.
Address 3: (1, 2). d = √(1 + 4) = √5 ≈ 2.24 km. ✓ Inside zone.
Address 4: (4, 5). d = √(16 + 25) = √41 ≈ 6.40 km. ✓ Inside zone.
All four distances (5, 10, 2.24, 6.40 km) are different, and address 1 (and 2) use a Pythagorean triple.
Marking: 1 mark per address with correct distance shown (up to 4 marks). Award full 4 if all conditions met. If a student lists addresses outside the zone or repeats a distance, deduct 1 mark per fault. At least one triple-based address is required for full marks.