Goal: Connect the five dots into a single continuous path so the consecutive segment lengths (in order) are 1, √10, 5, √5.
- Label the points for clarity: A = (5,0), B = (6,3), C = (1,3), D = (6,2), E = (3,2).
- Use the Pythagorean theorem in the coordinate plane. For any two points (x1,y1) and (x2,y2), squared distance = (x2-x1)^2 + (y2-y1)^2. Comparing squared distances avoids unnecessary square roots.
- Compute squared distances between useful pairs (showing only those needed):
- BD: (6,2) → (6,3): dx=0, dy=1 ⇒ 0^2+1^2 = 1 ⇒ distance = 1.
- BA: (6,3) → (5,0): dx=-1, dy=-3 ⇒ 1+9 = 10 ⇒ distance = √10.
- AC: (5,0) → (1,3): dx=-4, dy=3 ⇒ 16+9 = 25 ⇒ distance = 5.
- CE: (1,3) → (3,2): dx=2, dy=-1 ⇒ 4+1 = 5 ⇒ distance = √5.
- Match these to the required ordered lengths 1, √10, 5, √5. The chain D → B → A → C → E produces the needed sequence:
- D(6,2) → B(6,3): distance = 1
- B(6,3) → A(5,0): distance = √10
- A(5,0) → C(1,3): distance = 5
- C(1,3) → E(3,2): distance = √5
- Therefore the correct continuous path through all five marked dots is: D → B → A → C → E (that is, (6,2) → (6,3) → (5,0) → (1,3) → (3,2)).
Teacher note (ACARA v9 alignment): This task practises applying the Pythagorean theorem in coordinate geometry, developing procedural fluency and reasoning when computing distances—aligned to ACARA v9 expectations for Years 8–9 mathematical problem solving and proof.
50-word exemplar teacher comment (Amy Chua 'Tiger Mother' cadence, ACARA v9 aligned):
Excellent: you solved the Pythagorean Paths accurately, showing precise coordinate calculations, systematic checking of each segment length, and clear reasoning. Continue rigorous practice: compare squared distances, reduce errors, and justify choices. Meets ACARA v9 expectations for applying the Pythagorean theorem in coordinate contexts with procedural fluency and logical argumentation consistently.
If you want, I can draw the 7×7 grid and mark the path, or create a printable worksheet so you can practice similar puzzles step-by-step.