If AD, BE and CF are medians of `DeltaABC`, then which one of the following statements is correct ?

Options

A. `(AD +BE+CF) lt AB + BC + CA `
B. `AD+BE + CF gt AB + BC + CA `
C. `AD+BE+CF = AB + BC + CA `
D. `AD+BE+CF= sqrt(2) (AB+BC+CA) `

Answer

To solve the problem regarding the medians of triangle ABC, we will analyze the properties of medians and their relationship with the sides of the triangle.

### Step-by-Step Solution:

1. **Understanding Medians**:
A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In triangle ABC, AD, BE, and CF are medians, where D, E, and F are the midpoints of sides BC, CA, and AB respectively.

**Hint**: Remember that each median divides the opposite side into two equal segments.

2. **Properties of Medians**:
It is known that the sum of the lengths of the medians of a triangle is always greater than three-fourths of the sum of the lengths of the sides of the triangle. This can be expressed mathematically as:
\[ AD + BE + CF > \frac{3}{4} (AB + BC + CA) \]

**Hint**: Look for relationships between the medians and the sides of the triangle.

3. **Multiplying the Inequality**:
If we multiply the entire inequality by 4, we get:
\[ 4(AD + BE + CF) > 3(AB + BC + CA) \]

**Hint**: Multiplying both sides of an inequality by a positive number preserves the inequality.

4. **Conclusion**:
From the properties of medians, we can conclude that the sum of the medians is less than the sum of the sides. Therefore, the correct statement regarding the medians of triangle ABC is that the sum of the medians (AD + BE + CF) is less than the sum of the sides (AB + BC + CA).

**Hint**: Always check the relationships between different geometric properties to derive conclusions.

### Final Answer:
The correct statement is that the sum of the medians is less than the sum of the sides of the triangle.