Pythagorean Paths (Age 14): Build a path with lengths 1, √10, 5, √5 — ACARA v9 aligned solution

Step-by-step, ACARA v9 aligned solution for a 14-year-old to the Pythagorean Paths puzzle (Beast Academy Practice 5D Ch.11). Coordinates, Pythagorean checks and a clear path that matches distances 1, √10, 5, √5.

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Goal: Place five distinct grid points and join them in order so that the consecutive segment lengths are 1, √10, 5 and √5. I will show coordinates on an integer grid, compute each distance using the Pythagorean theorem, and verify the required list of lengths.

  1. Choose a starting point and place the five points (coordinates):
    • A = (0, 0)
    • B = (1, 0)
    • C = (4, 1)
    • D = (7, 5)
    • E = (5, 4)
    (All points are lattice points on the integer grid.)
  2. Compute vectors between consecutive points and verify lengths using the Pythagorean theorem:
    • Segment AB: vector = B - A = (1, 0).
      Squared length = 1^2 + 0^2 = 1 ⇒ length = √1 = 1 (matches required first length).
    • Segment BC: vector = C - B = (4 - 1, 1 - 0) = (3, 1).
      Squared length = 3^2 + 1^2 = 9 + 1 = 10 ⇒ length = √10 (matches required second length).
    • Segment CD: vector = D - C = (7 - 4, 5 - 1) = (3, 4).
      Squared length = 3^2 + 4^2 = 9 + 16 = 25 ⇒ length = √25 = 5 (matches required third length).
    • Segment DE: vector = E - D = (5 - 7, 4 - 5) = (-2, -1).
      Squared length = (-2)^2 + (-1)^2 = 4 + 1 = 5 ⇒ length = √5 (matches required fourth length).
  3. Summary of the path:

    Path A → B → C → D → E with coordinates:

    (0,0) → (1,0) → (4,1) → (7,5) → (5,4)

    Segment lengths (in order): 1, √10, 5, √5 — exactly as required.

  4. Why this works (connection to Pythagorean theorem):

    On an integer grid, the square of the distance between two points (x1,y1) and (x2,y2) is (x2-x1)^2 + (y2-y1)^2. For each segment above we computed that square and confirmed it equals 1, 10, 25, 5 respectively. Taking square roots gives the required lengths 1, √10, 5 and √5. That is a direct application of the Pythagorean theorem to right triangles formed by horizontal and vertical differences.

  5. Teacher feedback (straight, exacting — for exemplary performance):

    Good. You must always show the coordinates you chose, the vector differences, and the squared distances. That is how you demonstrate full understanding and earn full marks. This solution is neat: each step has explicit calculation, and the path uses only lattice points so your geometry is exact, not approximate.

    To improve further: practice finding alternate placements that satisfy the same sequence, and practice explaining why certain displacement pairs (like (3,1) for √10) are the only integer-vector choices on a lattice. Be precise, write each step, and never skip the Pythagorean check.

Final answer (compact): A = (0,0), B = (1,0), C = (4,1), D = (7,5), E = (5,4). Distances AB = 1, BC = √10, CD = 5, DE = √5. Verified by Pythagorean theorem.

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