Understanding and using the distance and midpoint formulas.

Distance Formula

The distance formula allows you to find the distance between any two points on a coordinate plane.

For points (x₁, y₁) and (x₂, y₂), the distance is:

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

This formula is derived from the Pythagorean theorem. If you draw a right triangle between two points:

Example: Using the Distance Formula

Let's find the distance between points (2, 3) and (5, 7):

$$d = \sqrt{(5-2)^2 + (7-3)^2}$$

$$d = \sqrt{3^2 + 4^2}$$

$$d = \sqrt{9 + 16}$$

$$d = \sqrt{25} = 5$$

The midpoint formula finds the coordinates of the point exactly halfway between two points.

For points (x₁, y₁) and (x₂, y₂), the midpoint is:

$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

The formula works by:

Let's find the midpoint between (2, 3) and (6, 9):

$$x = \frac{2 + 6}{2} = \frac{8}{2} = 4$$

$$y = \frac{3 + 9}{2} = \frac{12}{2} = 6$$

Therefore, the midpoint is (4, 6)

Finding the length of a line segment
Determining if a triangle is right, isosceles, or equilateral by comparing side lengths
Finding the center of a circle (midpoint of diameter)
Verifying if a point is equidistant from two other points
Finding coordinates of points that divide a line segment in a given ratio