Distance and Midpoint Formulas

The distance formula is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

The midpoint formula is: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ .

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

Using the distance formula: $$\sqrt{(7 - (-4))^2 + (y - 5)^2} = \sqrt{290}$$ . Solving gives y = 18.

Using the distance formula: $$\sqrt{(5 - 2)^2 + (y - 9)^2} = \sqrt{34}$$ . Solving gives y = 14.

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

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

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

Using the distance formula: $$d = \sqrt{(3 - 3)^2 + (8 - 4)^2} = \sqrt{0 + 16} = 4$$ .

Using the distance formula: $$d = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ .