Solving for the Distance Between the Two Furthest Points on a Line

When dealing with points on a line, it's important to understand how to calculate the distances between them. This involves both direct measurement and strategic addition of segment lengths. Given the problem of determining the distance between the two furthest points on a line with the following segment lengths:

AB 2 BC 6 CD 8 DE 10 EF 20 FA 22

Initial Calculation

One way to approach the problem is to add up all the given segment lengths to find the total distance between the two farthest points. Although this method might seem straightforward, it does not necessarily represent the correct configuration of the points on the line. Here’s how this calculation would look:

AB   BC   CD   DE   EF   FA  2   6   8   10   20   22  68

However, this suggests a linear path that might not be the configuration of the given points. It's essential to consider the actual configuration of the points.

Correct Configuration Analysis

To find the actual distance between the two furthest points, we need to consider the possible configurations of the points on the line. Let's assume the points are placed in the following order: A → B → C → D → E → F or a reverse order.

Let's break down the distances as follows:

A to B: 2 units B to C: 6 units C to D: 8 units D to E: 10 units E to F: 20 units F to A: 22 units

Given these distances, we can determine the total distance by considering the most direct path from one end to the other. There are a few possibilities:

Positive Direction Path

A → B → C → D → E → F Distance: 2 6 8 10 20 46

Negative Direction Path

F → A → B → C → D → E Distance: 22 - 2 - 6 - 8 - 10 4

Clearly, the positive direction path shows the true distance between the starting and ending points, which in this case is 46 units. However, the problem specifies a configuration that includes a reverse direction (FA 22) which points to a reversed order. This leads us to the most likely configuration, where the points are in reverse order A → F → E → D → C → B.

Reversed Order Configuration

Let's consider the reversed order:

A → F: 22 units F → E: -20 units (negative direction) E → D: 10 units (positive direction) D → C: -8 units (negative direction) C → B: -6 units (negative direction) B → A: 2 units (positive direction)

By adding the distances in the correct order, we get:

AF - AB   DE - CD - CB  22 - 2   10 - (-8) - (-6)  24   10   8   6  48

This calculation shows that the distance between the two furthest points, B and F, is actually 48 units.

In conclusion, the distance between the two furthest points on the line is 48 units, considering the given segment lengths and their directions.

Conclusion

Understanding the correct configuration of points on a line is crucial for solving such geometric problems accurately. By considering the positive and negative directions of segment lengths, we can determine the actual distance between the two furthest points. The distance between the furthest points B and F is 48 units.