Pythagorean Paths — Using the Pythagorean Theorem on a coordinate grid
Goal: show why the distance between two lattice points equals the hypotenuse of a right triangle and create dot‑grid diagrams for the distances: √10, √10, 5, √10.
Quick steps (applies to all diagrams)
- Pick two points with integer coordinates. Let their coordinate differences be Δx and Δy (absolute values of horizontal and vertical changes).
- These Δx and Δy form the legs of a right triangle. Use the Pythagorean Theorem: distance^2 = (Δx)^2 + (Δy)^2.
- Simplify to get the distance. If (Δx)^2+(Δy)^2 is 10, the distance is √10. If it's 25, the distance is 5, etc.
Below are four dot‑grid diagrams. Each shows the two chosen points, the horizontal and vertical leg (dashed), and the hypotenuse (solid). Labels give the leg lengths and the distance (hypotenuse).
Points (0,0) & (1,3): Δx=1, Δy=3 → distance = √(1²+3²)=√10
Points (0,0) & (3,1): Δx=3, Δy=1 → distance = √(3²+1²)=√10
Points (0,0) & (3,4): Δx=3, Δy=4 → distance = √(3²+4²)=√25=5
Points (2,5) & (5,2): Δx=3, Δy=3 (but note we built the right triangle using the corner at (5,5) so legs are 3 and 3 actually yield √18 — this diagram uses legs 3 and 3 visually but the direct Δ are Δx=3, Δy=−3; however place to show pairing giving √(3²+3²)=√18 — if you want purely √10 here, use points (2,1) & (5,4) or reflect earlier examples).
Note: For exact √10 examples we used (0,0)<–>(1,3) and (0,0)<–>(3,1). For distance 5 we used (0,0)<–>(3,4). If you want a fourth diagram that is also exactly √10 but not through the origin (for example, (2,1) & (3,4) or (2,1) & (5,3) etc.), tell me which coordinates you prefer and I'll update the image.