Mathematics • Year 10 • Unit 2 • Lesson 15
Distance and Midpoint in the Real World
Use the distance and midpoint formulas on real coordinate setups — map grids, halfway meeting points, drone delivery, walking paths and triangulation. Then explain why the distance formula is just Pythagoras dressed differently.
1. Word problems
Treat each map / grid as a coordinate plane (1 unit = 1 km unless told otherwise). Show all working.
1.1 — School to park. The school is at S(2, 3) and the park is at P(8, 11) on a town grid where 1 unit = 1 km.
(a) Find the straight-line distance from S to P (exact, then to 1 dp).
(b) Find the midpoint, where the bus stop should go. 3 marks
1.2 — Friends meet halfway. Mai lives at (−3, 4) and Sam lives at (7, −2) on a city grid (km).
(a) Find the midpoint M where they meet.
(b) Find Mai's walking distance from home to M (exact, then to 1 dp).
(c) Verify Sam's walking distance is the same. 4 marks
1.3 — Drone delivery. A drone leaves base at B(0, 0) and delivers to point D(5, 12) on a flat-ground grid (1 unit = 100 m).
(a) Find the straight-line flight distance in metres.
(b) If the drone's battery is enough for 1500 m, can it make the round trip? Show your numbers. 3 marks
1.4 — Path centroid. A walking trail joins three viewpoints A(2, 1), B(6, 1), C(4, 7). The "rest hut" should sit at the midpoint of the longest side.
(a) Find the length of each side AB, BC, AC.
(b) Identify the longest side and find the midpoint to place the rest hut. 4 marks
1.5 — Endpoint problem. The midpoint of a fence-line segment is at M(6, 3). One endpoint is at A(2, 8). Find the other endpoint B (the fence's far corner).
Show both x_B = 2x_M − x_A and y_B = 2y_M − y_A, then check by computing the midpoint of A and your B. 3 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 A classmate says: "The distance formula is just Pythagoras' theorem in disguise." Using the words horizontal run, vertical rise and hypotenuse, explain (i) why the classmate is correct, (ii) draw a quick sketch (in words is fine) showing the right triangle formed by two points on a coordinate plane, and (iii) confirm using the points (1, 2) and (4, 6): show the run, the rise, and apply the Pythagoras formula a² + b² = c² to get the same distance the formula gives.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — School to park
(a) Δx = 6, Δy = 8. d = √(36 + 64) = √100 = 10 km. (b) M = ((2+8)/2, (3+11)/2) = (5, 7).
1.2 — Halfway meet
(a) M = ((−3+7)/2, (4−2)/2) = (2, 1). (b) Mai: √((2−(−3))² + (1−4)²) = √(25 + 9) = √34 ≈ 5.8 km. (c) Sam: √((7−2)² + (−2−1)²) = √(25 + 9) = √34 ≈ 5.8 km ✓ — equal as expected.
1.3 — Drone delivery
(a) Δx = 5, Δy = 12. d = √(25 + 144) = √169 = 13 grid units = 1300 m. (b) Round trip = 2 × 1300 = 2600 m, which is more than the 1500 m range — no, the drone cannot make a round trip. It could deliver one-way (1300 m) but not return.
1.4 — Path centroid
(a) AB = √((6−2)² + 0²) = 4. BC = √((4−6)² + (7−1)²) = √(4 + 36) = √40 = 2√10. AC = √((4−2)² + (7−1)²) = √(4 + 36) = √40 = 2√10. (b) BC and AC tie at 2√10 ≈ 6.32 (both longest). Midpoint of BC: ((6+4)/2, (1+7)/2) = (5, 4). (Midpoint of AC: (3, 4) is a valid alternative answer if AC is chosen.)
1.5 — Endpoint
x_B = 2(6) − 2 = 10; y_B = 2(3) − 8 = −2. B = (10, −2). Check midpoint of A(2,8) and B(10,−2): ((2+10)/2, (8−2)/2) = (6, 3) ✓.
2.1 — Explain (sample response)
(i) The distance between two points on a coordinate plane forms the hypotenuse of a right triangle whose legs are the horizontal run (x₂ − x₁) and the vertical rise (y₂ − y₁). Pythagoras says hypotenuse² = leg₁² + leg₂², so distance = √(run² + rise²) — that is literally the distance formula. (ii) Sketch in words: from (1,2), draw a horizontal arrow right to (4,2), then a vertical arrow up to (4,6). The diagonal from (1,2) to (4,6) is the hypotenuse. (iii) Confirm: run = 4 − 1 = 3; rise = 6 − 2 = 4. Pythagoras: c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. Distance formula gives √((4−1)² + (6−2)²) = √(9 + 16) = 5 ✓ — same answer.
Marking: 1 for the right-triangle identification; 1 for using all three required words; 1 for a clear sketch description; 1 for the Pythagoras vs distance verification.