Sweet Geometry: A Pythagorean Paths Challenge
ACARA v9 alignment: use Pythagoras and the distance formula in the coordinate plane (Year 9-style reasoning about distance in two dimensions).
You are given five marked dots on a 7×7 grid: (5,0), (6,3), (1,3), (6,2), (3,2). Your delicious task is to join them into a single continuous path so that the distances between consecutive dots, in order along the path, are:
1, √10, 5, √5
Rules: Use each dot exactly once. The straight-line (Euclidean) distance between adjacent dots along your path must match the lengths above, in that order.
Hints — gently coaxed
- Recall: distance between (x1,y1) and (x2,y2) is √[(x2−x1)² + (y2−y1)²] (Pythagoras in coordinates).
- Make a neat table of distances between every pair of the five points — like tasting each ingredient by itself before you mix.
- Look for pairs that give exactly 1, exactly √10, exactly 5 and exactly √5. Which pair is the only one that is 1? That gives a necessary adjacency.
- Try building the path step by step: place the unique length-1 pair as your first segment, then find which point next gives √10 from there, then which gives 5, and finally √5.
- If you prefer a visual approach, draw the 7×7 grid, plot the points, and sketch candidate segments. Taste and adjust until the sequence fits.
Worked example for the method (not the full solution)
Example calculation: distance between (5,0) and (6,3) = √[(6−5)² + (3−0)²] = √(1 + 9) = √10. Do this for each pair.
When you have a path that uses all five dots with the four segment lengths in order, write a short justification: show your pairwise calculations and explain why no other ordering works (or why this is unique).
BONUS (Beast Academy Practice 5D, Challenge)
Same puzzle, same coordinates, a touch harder than the book: find the single continuous path on the 7×7 grid connecting the marked dots so consecutive distances are 1, √10, 5, √5.
Teacher comments and exemplar rubric (50 words)
Delicious reasoning: students must order the five dots so segment lengths are 1, √10, 5, √5. Criteria: correct sequence, explicit use of distance formula or Pythagoras, calculations shown, neat coordinate labels, reflection on uniqueness. Exemplar path: (6,2)→(6,3)→(5,0)→(1,3)→(3,2). Praise careful algebra, tidy diagrams, and graceful written justification.