The distance formula

A quick reminder. A point (x, y) is x units from the y-axis and y units from the x-axis. A point on the x-axis is (x, 0), and a point on the y-axis is (0, y).

Distance along an axis is easy. A(4, 0) and B(6, 0) are 6 – 4 = 2 units apart. C(0, 3) and D(0, 8) are 5 units apart.

Any two points: make a right triangle. Join P(x₁, y₁) and Q(x₂, y₂). Draw a horizontal line from one and a vertical line from the other until they meet. Now PQ is the hypotenuse of a right triangle:

• the across side is the difference of the x's: x₂ – x₁
• the up side is the difference of the y's: y₂ – y₁

Pythagoras gives the distance formula:

PQ = √((x₂ – x₁)² + (y₂ – y₁)²)

Say it as a routine: subtract the x's and square; subtract the y's and square; add; take the root. A distance is never negative, so take only the positive root.

From the origin O(0, 0) the formula is simply OP = √(x² + y²).

Town B is 36 km east and 15 km north of town A. The straight-line distance is √(36² + 15²) = √(1296 + 225) = √1521 = 39 km.

Two traps: (–5)² = +25, so brackets matter. And √(a² + b²) is not a + b: √(9 + 16) = 5, not 7.