Goal
Use the Pythagorean Theorem to find the straight-line distance between two points on a dot-grid (coordinate plane). We'll make four diagrams where the distances are: √10, √10, 5, and √10.
Step-by-step method (what to do every time)
- Write the coordinates of the two points: (x1, y1) and (x2, y2).
- Find the horizontal difference dx = |x2 − x1| and the vertical difference dy = |y2 − y1|.
- Use the Pythagorean Theorem: distance = sqrt(dx^2 + dy^2). This works because dx and dy are the legs of a right triangle and the distance is the hypotenuse.
- Simplify the square root if possible (for example, sqrt(25) = 5).
Diagrams (dot-grid) with calculations
Each diagram below shows a small dot-grid (x and y are integers). The bold points are the two points whose distance we find. The right triangle is drawn so you can see dx and dy.
Calculation: dx = |1−0| = 1, dy = |3−0| = 3. Distance = √(1^2 + 3^2) = √(1 + 9) = √10.
Calculation: dx = |3−0| = 3, dy = |1−0| = 1. Distance = √(3^2 + 1^2) = √(9 + 1) = √10.
Calculation: dx = |3−0| = 3, dy = |4−0| = 4. Distance = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Calculation: dx = |3−2| = 1, dy = |5−2| = 3. Distance = √(1^2 + 3^2) = √(1 + 9) = √10.
Quick tips:
- If dx and dy are integers, you often get familiar square-root results (like 5 from a 3-4-5 triangle).
- Look for small integer dx,dy pairs that satisfy dx^2 + dy^2 = N when you want to make a distance √N. For √10, try (1,3) or (3,1).
- Always label the coordinates and check dx, dy carefully — sign doesn't matter because you take absolute differences.
Related standard: 8.G.B.8 — applying the Pythagorean Theorem to find distances in the coordinate plane.