Distance formula rule

Distance formula rule, formula for the distance d. The distance is always taken to be positive, and it is not a directed distance from A to B. When A and B do not lie on the same horizontal or vertical line.

If A and B lie on a line parallel to one of the coordinate Axis. Then, by the formula 2,The distance AB is the absolute value of the directed distance → AB

Distance formula Proof:

Let A(x ₁, y ₁)and B(x ₂, y ₂) be two point in the plane. We can find distance  from the right triangle AQB by using the pathagoras theorem, we have

Distance formula rule

d² = AB² = A Q² + QB² →→→(1)

 ∣A Q∣ = ∣RS∣ = ∣RO+OS∣

              ∣RS∣=∣-OR+OS∣
                     = ∣x ₁-x ₁∣

∣QB∣= ∣SB-SQ∣=∣OM-ON∣
                         = ∣y ₁-y ₁∣

therefore, (1) can be written in the form

d²= d=(x ₂- x ₁)²+(y ₂- y ₁)²

d= ∣A B∣ =√(x ₂- x ₁)²+(y ₂- y ₁)  ………….(2)

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