Distance Formula

The distance formula is used to determine the distance between two points in a coordinate plane. It is given by: $$d = \text{sqrt}((x_2 - x_1)^2 + (y_2 - y_1)^2)$$ .

Using the distance formula: $$d = \text{sqrt}((8 - (-3))^2 + (-42 - (-11))^2) = \text{sqrt}(11^2 + (-31)^2) = \text{sqrt}(121 + 961) = \text{sqrt}(1082) \ d \approx 32.9$$ .

Using the distance formula: $$d = \text{sqrt}((3 - 5)^2 + (-2 - 0)^2) = \text{sqrt}((-2)^2 + (-2)^2) = \text{sqrt}(4 + 4) = \text{sqrt}(8) \ d \approx 2.83$$ .

Using the distance formula: $$d = \text{sqrt}((3 - 4)^2 + (4 - 2)^2) = \text{sqrt}((-1)^2 + (2)^2) = \text{sqrt}(1 + 4) = \text{sqrt}(5) \ d \approx 2.24$$ .

Using the distance formula: $$d = \text{sqrt}((5 - 2)^2 + (7 - 3)^2) = \text{sqrt}(3^2 + 4^2) = \text{sqrt}(9 + 16) = \text{sqrt}(25) = 5$$ .

The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The distance between two identical points is 0.

The distance is undefined if the points are not defined in the coordinate system, such as if one or both points are not real numbers.

The distance is 0, as both points are the same.

The distance is expressed as: $$d = \text{sqrt}((x_2 - x_1)^2 + (y_2 - y_1)^2)$$ .