NIMCET 2008 — Mathematics PYQ

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 $\Delta ABD$

In $\Delta ABD$, the sum of any two sides must be greater than the third side:

Since $BD=BM+MD$, we can rewrite this as:

3. Apply Triangle Inequality to $\Delta MDC$

Similarly, in $\Delta MDC$, we have:

4. Combine the Inequalities

Add Equation 1 and Equation 2 together:

Simplify both sides by cancelling out $MD$ (which appears on both sides):

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)