Midpoint and Distance between Two Points

Understanding Midpoints

The midpoint of a line segment is the point that divides the segment into two equal parts. If you have two points, $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $(M)$ is calculated using the formula:

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

This formula gives you the average of the x-coordinates and the average of the y-coordinates.

Understanding Distance

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be found using the distance formula:

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

This formula is derived from the Pythagorean theorem and calculates the length of the line segment connecting the two points.

Examples

Example 1: Finding the Midpoint

Find the midpoint of the points $(2, 3)$ and $(4, 7)$.

Example 2: Finding the Distance

Calculate the distance between the points $(1, 2)$ and $(4, 6)$.