# An Odyssey Through Distance and Midpoint Formulas in the Plane

Navigating the intricate tapestry of coordinate geometry, scholars and mathematicians have conjured formulas that effortlessly unveil the distance and the central point, or midpoint, between any two distinct points on the plane. These calculations, seemingly straightforward, can be understood as manifestations of the time-honored Pythagorean theorem, artfully extended to a two-dimensional realm. Let's embark on a detailed odyssey to understand and harness these formulas. ## Step-by-step Guide to Understanding Distance and Midpoint Formulas

Here is a step-by-step guide to understanding distance and midpoint formulas:

### Distance Formula:

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

Given: Points $$A \ (x_1​,y_1​)$$ and $$B \ (x_2​,y_2​)$$

Formula: $$d=\sqrt{(x_2​−x_1​)^2+(y_2​−y_1​)^2​}$$

#### Steps to Find the Distance:

1. Identify the Coordinates: First, determine the coordinates of the two points.
2. Substitute into the Formula:
• Subtract the $$x$$-coordinates: $$x_2​−x_1​$$
• Subtract the $$y$$-coordinates: $$y_2​−y_1​$$
3. Square the Results: Square the results of the above subtractions.
5. Square Root: Take the square root of the sum to get the distance.

### Midpoint Formula:

The midpoint formula is used to determine the midpoint (or the average) of the segment connecting two points in a coordinate plane.

Given: Points $$A \ (x_1​,y_1​)$$ and $$B \ (x_2​,y_2​)$$

Formula: $$M=$$$$(\frac{x_1​+x_2}{2}$$​​,$$\frac{y_1​+y_2}{2})$$

#### Steps to Find the Midpoint:

1. Identify the Coordinates: First, determine the coordinates of the two points.
2. Average the $$x$$-coordinates: Add the $$x$$-coordinates and divide by $$2$$: $$\frac{x_1​+x_2}{2}$$
3. Average the $$y$$-coordinates: Add the $$y$$-coordinates and divide by $$2$$: $$\frac{y_1​+y_2}{2})\​​ 4. Write the Midpoint: Combine the averaged \(x$$ and $$y$$ coordinates to get the midpoint: $$(\frac{x_1​+x_2}{2}$$​​,$$\frac{y_1​+y_2}{2})$$

### Examples:

Example 1:

Find the distance between two points $$A \ (2,4)$$ and $$B \ (6,8)$$.

Solution:

$$d=\sqrt{(6−2)^2+(8−4)^2​}$$

$$d=\sqrt{4^2+4^2​}$$

$$d=\sqrt{32}​$$

$$d=4 \sqrt{2}​$$

Example 2:

Find the midpoint of two points, $$A \ (−3,2)$$ and $$B \ (5,8)$$, on the coordinate plane.

Solution:

To find the midpoint, $$M$$, of the segment connecting points $$A$$ and $$B$$.

Using the Midpoint Formula:

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

1. Average the $$x$$-coordinates: $$x_m​=\frac{−3+5}{2}​=1$$
2. Average the $$y$$-coordinates: $$y_m​=\frac{2+8}{2}​=5$$
3. Combine the Results: Thus, the midpoint $$M$$ is $$(1,5)$$.

The midpoint between the points $$A \ (−3,2)$$ and $$B\ (5,8)$$ is $$M \ (1,5)$$.

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