Distance and Midpoint Formula Review

The distance formula is given by $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ , which calculates the distance between two points (x1, y1) and (x2, y2) in a Cartesian plane.

The midpoint formula is given by $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ , which finds the midpoint M of the line segment connecting points (x1, y1) and (x2, y2).

Using the distance formula: $$d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ .

Using the midpoint formula: $$M = \left(\frac{2 + 4}{2}, \frac{3 + 7}{2}\right) = (3, 5)$$ .

Using the distance formula: $$d = \sqrt{(2 - (-1))^2 + (2 - (-1))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{18} = 3\sqrt{2}$$ .

The distance formula is used to determine the length of a line segment between two points in a coordinate plane, which is fundamental in geometry.

Using the distance formula: $$5 = \sqrt{(x - 1)^2 + (5 - 2)^2} \Rightarrow 25 = (x - 1)^2 + 9 \Rightarrow (x - 1)^2 = 16 \Rightarrow x - 1 = 4 \Rightarrow x = 5 \text{ or } x - 1 = -4 \Rightarrow x = -3$$ .

Using the distance formula: $$d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ .

The length of a diagonal can be found using the distance formula between two opposite corners of the rectangle.

Using the distance formula: $$d = \sqrt{(1 - 1)^2 + (5 - 2)^2} = \sqrt{0 + 9} = 3$$ .