Distance between two points

Easy case first. If two points lie on a line parallel to an axis, only one coordinate changes. The distance is just the difference of that coordinate. W1W2 and S1S2 in Fig. 1.5 are like that.

"What if the line is slanted?" Take A (3, 4) and D (7, 1). To get from A to D you move:

• 7 – 3 = 4 units across (along x);
• 4 – 1 = 3 units down (along y).

Those two moves are the two short sides of a right triangle (right angle at C), and AD is the longest side, the hypotenuse. By the Baudhāyana–Pythagoras theorem from Grade 8: AD = √(4² + 3²) = √25 = 5 units.

The same way:

• DM = √(2² + 5²) = √29 units;
• MA = √(6² + 2²) = √40 units.

The general rule. The distance between (x1, y1) and (x2, y2) is √((x2 – x1)² + (y2 – y1)²). In words: square the move across, square the move up or down, add, and take the square root.

"Does it matter which point I start from?" No. A move of –4 or +4 gives the same 16 when squared. We only measure how big the shift is.