An Odyssey Through Distance and Midpoint Formulas in the Plane

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.
4. Add Them Together: Add the squared results together.
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 \(M \ (1,5)\).