Understanding the Problem
To prove that if points A, B, and C are collinear (on the same line) with B between A and C, then the relationship AB + BC = AC holds true.
Defining the Distances
- Let AB be the distance from point A to point B.
- Let BC be the distance from point B to point C.
- Let AC be the total distance from point A to point C.
Visual Representation
Imagine a straight line where:
- A is on the left,
- B is in the middle,
- C is on the right.
This can be represented as:
A ---- B ---- C
Breaking Down the Distances
1. Distance AB: This is the length between points A and B.
2. Distance BC: This is the length between points B and C.
3. Distance AC: This is the overall length from point A to point C.
Combining the Distances
When we move from A to C, we first travel from A to B and then from B to C.
- Therefore, the total distance AC can be expressed as the sum of the two distances:
AC = AB + BC.
Conclusion
This demonstrates that the sum of the segments AB and BC is equal to the segment AC when point B lies between points A and C. Thus, we conclude that:
AB + BC = AC.
This property is a fundamental aspect of linear measurement and holds true for any points on a straight line.