Mastering the Midpoint & Distance on the Number Line

Step-by-step Guide: Distance and Midpoint on the Number Line

Understanding the Midpoint:
The midpoint between two numbers (or points) on a number line is the number that lies exactly halfway between them. The formula to find the midpoint \(M\) between two points \(A(x_1)\) and \(B(x_2)\) is:
\(M = \frac{x_1 + x_2}{2} \)

the Concept of Distance:
The distance between two numbers (or points) on a number line is the absolute difference between them. The formula to find the distance \(D\) between two points \(A(x_1)\) and \(B(x_2)\) is:
\( D = |x_2 – x_1| \)

Examples

Example 1: Midpoint
Find the midpoint between the numbers \(4\) and \(10\) on the number line.

Solution:
Using the midpoint formula:
\( M = \frac{4 + 10}{2} \)
\( M = \frac{14}{2} \)
\( M = 7 \)
So, the midpoint is \(7\).

Example 2: Distance
Find the distance between the numbers \(-3\) and \(5\) on the number line.

Solution:
Using the distance formula:
\( D = |5 – (-3)| \)
\( D = |5 + 3| \)
\( D = 8 \)
So, the distance is \(8\) units.

Practice Questions:

1. Find the midpoint between the numbers \(2\) and \(8\) on the number line.
2. Calculate the distance between the numbers \(-7\) and \(3\) on the number line.
3. If the midpoint between two numbers \(a\) and \(b\) is \(5\), what is the value of \(b\) given \(a = 3\)?

Answers:

1. The midpoint is \(5\).
2. The distance is \(10\) units.
3. The value of \(b\) is \(7\).