Explanation
1. Construct a Geometric Extension
To prove the relationship between the outer sides and internal segments, let's extend the line segment BM to intersect the side AC at a point, say D.
2. Apply Triangle Inequality to ΔABD
In ΔABD, the sum of any two sides must be greater than the third side:
Since BD=BM+MD, we can rewrite this as:
AB + AD > BM + MD \quad \dots \text{(Equation 1)}
3. Apply Triangle Inequality to ΔMDC
Similarly, in ΔMDC, we have:
MD + DC > MC \quad \dots \text{(Equation 2)}
4. Combine the Inequalities
Add Equation 1 and Equation 2 together:
(AB + AD) + (MD + DC) > (BM + MD) + MC
Simplify both sides by cancelling out MD (which appears on both sides):
AB + (AD + DC) > BM + MC
Since AD+DC is simply the length of side AC:
Conclusion
For any point M inside a triangle ABC, the sum of the two sides of the triangle is always greater than the sum of the segments connecting the internal point to the endpoints of the third side.
Correct Option: (b)
