Read the statemenst carefully and state 'T' for true and 'F' for false .
1. If a line divides any two sides of a triangle in the same ratio , then the line is parallel to the third side of the triangle .
2 . The internal bisector of an angle of a triangle divides the opposite side inernally in the ratio of the sides containing the angle .
3 . If a line through one vertex of a triangle divides the opposite in the ratio of other two sides , then the line bisects the angle at the vertex .
4.Any line parallel to the parallel sides dividesproportionally .
5. Two times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle .

Options

A. `{:(1,2,3,4,5),(T,T,T,T,T):}`
B. `{:(1,2,3,4,5),(T,T,T,T,F):}`
C. `{:(1,2,3,4,5),(F,T,F,T,F):}`
D. `{:(1,2,3,4,5),(T,T,F,T,F):}`

Answer

To solve the problem, we will analyze each statement one by one and determine whether it is true (T) or false (F).

### Step-by-Step Solution:

1. **Statement 1**: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side of the triangle.
- **Analysis**: This statement is true based on the Basic Proportionality Theorem (also known as Thales' theorem). If a line divides two sides of a triangle proportionally, it is parallel to the third side.
- **Conclusion**: T

2. **Statement 2**: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
- **Analysis**: This statement is true according to the Angle Bisector Theorem, which states that the angle bisector divides the opposite side in the ratio of the other two sides.
- **Conclusion**: T

3. **Statement 3**: If a line through one vertex of a triangle divides the opposite side in the ratio of the other two sides, then the line bisects the angle at the vertex.
- **Analysis**: This statement is true and is known as the Converse of the Angle Bisector Theorem. If a line divides the opposite side in the ratio of the other two sides, it must be the angle bisector.
- **Conclusion**: T

4. **Statement 4**: Any line parallel to the parallel sides divides proportionally.
- **Analysis**: This statement is true. If a line is parallel to one side of a triangle, it divides the other two sides proportionally (Basic Proportionality Theorem).
- **Conclusion**: T

5. **Statement 5**: Two times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.
- **Analysis**: This statement is false. The relationship described does not hold true mathematically.
- **Conclusion**: F

### Final Answers:
1. T
2. T
3. T
4. T
5. F