Distance Formula

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

Using the Distance Formula: $$d = \text{sqrt}((3 - 8)^2 + (-4 - 8)^2) = \text{sqrt}(25 + 144) = \text{sqrt}(169) = 13$$ .

Using the Distance Formula: $$d = \text{sqrt}((7 - (-4))^2 + (18 - 5)^2) = \text{sqrt}(121 + 169) = \text{sqrt}(290) \text{ (approximately 17.03)}$$ .

The correct expression is: $$d = \text{sqrt}((-2 - 3)^2 + (-1 - 1)^2)$$ .

Using the Distance Formula: $$d = \text{sqrt}((7 - 3)^2 + (56 - 24)^2) = \text{sqrt}(16 + 1024) = \text{sqrt}(1040) \text{ (approximately 32.2)}$$ .

Using the Distance Formula: $$d = \text{sqrt}((5 - 1)^2 + (6 - 3)^2) = \text{sqrt}(16 + 9) = \text{sqrt}(25) = 5$$ .

A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis). Each point on the plane is defined by an ordered pair (x, y).

The square root in the Distance Formula ensures that the distance is a non-negative value, as distance cannot be negative.

In geometry, distance refers to the length of the shortest path between two points, which can be calculated using various formulas depending on the context.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The Distance Formula is derived from this theorem.