Beast Academy 5D — Pythagorean Paths dot-grid diagrams for √10, √10, 5, √10 (Age 25)

Pythagorean Paths — visualizing distances on a dot grid

Below are four dot‑grid diagrams. Each shows a line segment between two lattice points (points with integer coordinates). I chose integer coordinate pairs whose differences produce the lengths you requested. For each diagram I show the right triangle formed by horizontal and vertical legs, label Δx and Δy, and show the computation that uses the Pythagorean Theorem to get the segment length.

1. Diagram A — length √10: endpoints (0,0) and (1,3). Δx=1, Δy=3 ⇒ distance = √(1²+3²)=√10.
2. Diagram B — length √10: endpoints (0,0) and (3,1). Δx=3, Δy=1 ⇒ distance = √(3²+1²)=√10.
3. Diagram C — length 5: endpoints (0,0) and (3,4). Δx=3, Δy=4 ⇒ distance = √(3²+4²)=5.
4. Diagram D — length √10: endpoints (4,1) and (5,4). Δx=1, Δy=3 ⇒ distance = √(1²+3²)=√10 (this one is shifted, showing the same leg pattern in another location).

Tip: any pair of integer differences (a,b) with a²+b²=10 (for √10) or a²+b²=25 (for 5) will work. For √10 we use (1,3) or (3,1). For 5 we use (3,4) or (4,3).

If you prefer, you can save any diagram as a PNG using the download buttons. Each download button converts the inline SVG to a PNG on your computer.

Quick step‑by‑step for any two lattice points (x1,y1) and (x2,y2):

1. Compute Δx = |x2 − x1| and Δy = |y2 − y1| (both integers for lattice points).
2. Form the right triangle with legs Δx and Δy; the distance between the points is the hypotenuse.
3. Use the Pythagorean Theorem: distance = √(Δx² + Δy²).

If you want different placements, different sizes, or pure PNG files sent directly, tell me the coordinate pairs or the exact image size and I can prepare new diagrams.