Question

a segment is m unites long. find the distance between the midpoints of the first and last parts in each case.
the segment is divided into 5 equal parts.

Answer 1

To find the distance between the midpoints of the first and last parts in a segment that is \( m \) units long, divided into 5 equal parts, follow these steps:

1. Calculate the length of each part:
\[
\text{Length of each part} = \frac{m}{5}
\]

2. Determine the midpoint of the first part:
- The first part extends from 0 to \( \frac{m}{5} \).
- The midpoint of the first part is:
\[
\text{Midpoint of the first part} = \frac{0 + \frac{m}{5}}{2} = \frac{m}{10}
\]

3. Determine the midpoint of the last part:
- The last part extends from \( 4 \cdot \frac{m}{5} \) to \( m \).
- The midpoint of the last part is:
\[
\text{Midpoint of the last part} = \frac{4\cdot\frac{m}{5} + m}{2} = \frac{4\cdot\frac{m}{5} + \frac{5m}{5}}{2} = \frac{\frac{4m + 5m}{5}}{2} = \frac{\frac{9m}{5}}{2} = \frac{9m}{10}
\]

4. Finally, calculate the distance between the two midpoints:
\[
\text{Distance} = \text{Midpoint of the last part} - \text{Midpoint of the first part} = \frac{9m}{10} - \frac{m}{10} = \frac{8m}{10} = \frac{4m}{5}
\]

Thus, the distance between the midpoints of the first and last parts is \( \frac{4m}{5} \) units.