Distance Formula

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

Using the Distance Formula: $$d = \sqrt{(3 - 8)^2 + (-4 - 8)^2} = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ .

Using the Distance Formula: $$d = \sqrt{(-4 - (-6))^2 + (-8 - (-3))^2} = \sqrt{(2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.3$$ .

The distance is represented as: $$d = \sqrt{(3 - 5)^2 + (-2 - 0)^2}$$ .

Using the Distance Formula: $$d = \sqrt{(6 - (-2))^2 + (5 - (-1))^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$$ .

Using the Distance Formula: $$d = \sqrt{(2 - 2)^2 + (-6 - (-4))^2} = \sqrt{(0)^2 + (-2)^2} = \sqrt{0 + 4} = \sqrt{4} = 2$$ .

In the Distance Formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ , \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are the coordinates of the second point.

The Distance Formula is significant as it allows us to calculate the straight-line distance between two points, which is fundamental in geometry, physics, and various applications.

Yes, the Distance Formula can be extended to three dimensions as: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ .

Using the Distance Formula: $$d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ .