Concept 7.1
The Distance Formula
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
The distance between P(x₁, y₁) and Q(x₂, y₂) is PQ = √((x₂ − x₁)² + (y₂ − y₁)²).
The intuition
Draw a horizontal line from P and a vertical line from Q. They meet at a right angle, and the distance PQ is the hypotenuse.
The two legs are just the differences in x and in y. So the distance formula is Pythagoras, with coordinates doing the measuring.
The attention
Look at the differences: they are the legs of a right triangle.
PQ = √( (x₂ − x₁)² + (y₂ − y₁)² ) not √( (x₁ + x₂)² + … )
Look hereNot here
- Look here
- The differences: the legs of a right triangle. x₂ − x₁ is the horizontal leg, y₂ − y₁ the vertical one. Pythagoras does the rest.
- Not here
- Sums of coordinates. Adding coordinates measures nothing on the picture. Distances come from differences.
- What it teaches
- Turn a formula back into its picture and it stops being something to memorise.