Goal
Apply the Pythagorean Theorem (distance formula) to find the distance between two points on a coordinate/dot grid. We'll build three dot-grid diagrams whose segment lengths are √10, √17, and √8 and show the step-by-step calculations.
Key idea (distance formula)
If two points are (x1,y1) and (x2,y2), then the distance d between them is
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
This is just the Pythagorean Theorem: the horizontal difference |x2 - x1| is one leg, the vertical difference |y2 - y1| is the other leg, and the distance is the hypotenuse.
Examples and dot-grid diagrams
For a clear dot-grid, I use integer-coordinate points so the legs are integers. The three choices below produce the desired distances because:
- √10 comes from legs 1 and 3 because 1^2 + 3^2 = 1 + 9 = 10.
- √17 comes from legs 1 and 4 because 1^2 + 4^2 = 1 + 16 = 17.
- √8 comes from legs 2 and 2 because 2^2 + 2^2 = 4 + 4 = 8.
Each diagram shows a 6×6 dot grid (points (0,0) through (5,5)). The origin is at (0,0). Points are plotted and the right triangle showing the legs and hypotenuse is drawn.
Choose endpoints (0,0) and (1,3). Horizontal leg length = |1-0| = 1. Vertical leg length = |3-0| = 3.
Distance = sqrt(1^2 + 3^2) = sqrt(10).
Choose endpoints (0,0) and (1,4). Horizontal leg length = 1. Vertical leg length = 4.
Distance = sqrt(1^2 + 4^2) = sqrt(17).
Choose endpoints (0,0) and (2,2). Horizontal leg length = 2. Vertical leg length = 2.
Distance = sqrt(2^2 + 2^2) = sqrt(8) = 2√2.
How to make your own dot-grid diagrams
- Pick two integer-coordinate points (x1,y1) and (x2,y2) so the differences dx = |x2-x1| and dy = |y2-y1| are integers you want.
- Compute d = sqrt(dx^2 + dy^2). That gives you the segment length you will see on the grid.
- Plot points on the dot grid, draw the horizontal and vertical legs, then draw the hypotenuse to show d visually.
Practice connection
This matches the Beast Academy 5D Pythagorean Paths enrichment: use the distance formula on coordinate points and visualize right triangles. Try the practice problems (5D pages 58–60, #142–155; #151–155 optional) and use these grid constructions to check answers visually.
If you want more diagrams (different orientations or non-integer endpoints), or printable PDFs of these grids, tell me which sizes or which distances you want and I will produce them.