Question

A segment is m units long. Find the distance between the midpoints of the first and last parts In each case. The segment is divided into three equal parts.

Answer 1

To divide a segment of length \( m \) into three equal parts, we can determine the length of each part. Each part will then have a length of:

\[
\text{length of each part} = \frac{m}{3}
\]

Given the segment is divided into three equal parts, we can denote the endpoints of the segment as \( A \), \( B \), and \( C \) such that:

- \( A \) is the starting point (0),
- \( B \) is the first endpoint (at \( \frac{m}{3} \)),
- \( C \) is the second endpoint (at \( \frac{2m}{3} \)),
- and the last endpoint (D) represents the end of the segment (at \( m \)).

Next, we can calculate the midpoints of the first and last parts:

1. **Midpoint of the first part (from A to B)**:
The midpoint \( M_1 \) of the first segment \( AB \) can be calculated as follows:

\[
M_1 = \frac{A + B}{2} = \frac{0 + \frac{m}{3}}{2} = \frac{\frac{m}{3}}{2} = \frac{m}{6}
\]

2. **Midpoint of the last part (from C to D)**:
The midpoint \( M_2 \) of the last segment \( CD \) can be calculated as follows:

\[
M_2 = \frac{C + D}{2} = \frac{\frac{2m}{3} + m}{2} = \frac{\frac{2m}{3} + \frac{3m}{3}}{2} = \frac{\frac{5m}{3}}{2} = \frac{5m}{6}
\]

Now we determine the distance \( d \) between the midpoints \( M_1 \) (of the first part) and \( M_2 \) (of the last part):

\[
d = M_2 - M_1 = \frac{5m}{6} - \frac{m}{6} = \frac{5m - m}{6} = \frac{4m}{6} = \frac{2m}{3}
\]

Thus, the distance between the midpoints of the first and last parts is:

\[
\boxed{\frac{2m}{3}}
\]