The distance between P(x1, y1) and Q(x2, y2) is :

Question

Accepted Answer

The distance between two points $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ in a two-dimensional Cartesian coordinate system is calculated using the distance formula. This formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. The distance formula is given by:

$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

**Explanation for Option 1:**

Option 1 correctly represents the distance formula. It calculates the horizontal distance between the x-coordinates, $ (x_2 - x_1) $, and the vertical distance between the y-coordinates, $ (y_2 - y_1) $. By squaring these differences, summing them, and then taking the square root, we obtain the straight-line distance between the two points, which is the hypotenuse of the right triangle formed by these differences. This is the standard and universally accepted method for calculating the distance between two points in a plane.

**Explanation for why other options are incorrect:**

- **Option 2: $\sqrt{(x_2 + x_1)^2 - (y_2 + y_1)^2}$**

This option incorrectly uses addition instead of subtraction for both the x and y coordinates. The distance formula requires the difference between the coordinates to reflect the actual distance along each axis. Additionally, subtracting the squares of the sums does not relate to the Pythagorean theorem and does not yield a meaningful geometric distance.

- **Option 3: $\sqrt{(x_2 - y_2)^2 + (x_1 - y_1)^2}$**

This option incorrectly mixes x and y coordinates in the differences. The distance formula requires comparing x-coordinates with x-coordinates and y-coordinates with y-coordinates to form the sides of the right triangle. Mixing them does not represent any meaningful geometric relationship and does not calculate the distance between the points.

- **Option 4: $\sqrt{(x_1 - x_2)^2 + (y_1 + y_2)^2}$**

While this option correctly uses the difference for the x-coordinates, it incorrectly uses addition for the y-coordinates. The distance formula requires the difference between the y-coordinates to accurately reflect the vertical distance. Using addition here does not correspond to the Pythagorean theorem and results in an incorrect calculation of distance.

In conclusion, Option 1 is the correct representation of the distance formula, accurately reflecting the geometric principles needed to calculate the distance between two points in a plane.