The distance d between two points P₁ = (x1, y1) and P₂ = (x2, y2) is given by the distance formula derived from the Pythagorean theorem. The formula is:
[tex]\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \][/tex]
This represents the straight-line distance in a Cartesian coordinate system.
Here's a step-by-step explanation of how to use this formula:
1. Subtract the x-coordinate of P₁ from the x-coordinate of P₂ to find the change in x (horizontal change), which is [tex]\( x2 - x1 \)[/tex].
2. Subtract the y-coordinate of P₁ from the y-coordinate of P₂ to find the change in y (vertical change), which is [tex]\( y2 - y1 \)[/tex].
3. Square each of the differences you found; that is, compute [tex]\( (x2 - x1)^2 \)[/tex] and [tex]\( (y2 - y1)^2 \)[/tex].
4. Sum the squared differences; add [tex]\( (x2 - x1)^2 \)[/tex] and [tex]\( (y2 - y1)^2 \)[/tex] together.
5. Take the square root of the sum you found in the previous step; calculate [tex]\( \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \)[/tex].
This result is the distance d between the two points P₁ and P₂.
