Introduction:

The distance formula in coordinate geometry is used to find the distance between two points in a coordinate plane. It is derived using the Pythagorean theorem and the concept of the right triangle. The formula can be applied in both two-dimensional (2D) and three-dimensional (3D) spaces.

Derivation:

To derive the distance formula, consider two points A(x1, y1) and B(x2, y2) in a 2D coordinate plane.

Step 1: Finding the length of the horizontal side:
- The horizontal side of the triangle formed by the two points has a length of |x2 - x1|, which can be obtained by subtracting the x-coordinates of the two points.

Step 2: Finding the length of the vertical side:
- The vertical side of the triangle has a length of |y2 - y1|, obtained by subtracting the y-coordinates of the two points.

Step 3: Applying the Pythagorean theorem:
- The distance between the two points is the hypotenuse of the right triangle formed by the horizontal and vertical sides. Using the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- By substituting the lengths of the horizontal and vertical sides, we get (|x2 - x1|)^2 + (|y2 - y1|)^2 = (distance)^2.

Step 4: Simplifying the equation:
- Simplify the equation by removing the absolute value signs since distances are always positive.
- We get (x2 - x1)^2 + (y2 - y1)^2 = (distance)^2.

Step 5: Solving for the distance:
- Take the square root of both sides of the equation to solve for the distance.
- The distance between the two points A(x1, y1) and B(x2, y2) is given by the formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Example:
Consider two points A(2, 3) and B(5, 7).
Using the distance formula, we can find the distance between these two points:
distance = sqrt((5 - 2)^2 + (7 - 3)^2)
distance = sqrt(3^2 + 4^2)
distance = sqrt(9 + 16)
distance = sqrt(25)
distance = 5

Conclusion:
The distance formula in coordinate geometry allows us to find the distance between two points in a coordinate plane. By using the Pythagorean theorem and the concept of right triangles, we can derive and apply the formula.