Question

The corners and mid-points of the sides of a triangle are named using the distinct letters P, Q, R, S, T, and U, but not necessarily in the same order. Consider the following statements:


The line joining P and R is parallel to the line joining Q and S.
P is placed on the side opposite to the corner T.
S and U cannot be placed on the same side.
Which one of the following statements is correct based on the above information?

• A. P cannot be placed at a corner
• B. S cannot be placed at a corner
• C. U cannot be placed at a mid-point
• D. R cannot be placed at a corner

Hint:

When analyzing geometric placement problems, pay close attention to parallelism, placement restrictions, and the geometric relationships between points and sides.

Answer 1

The Correct Option is B

Step 1: Understand the given conditions.
We are given that P, Q, R, S, T, and U are placed at various points of the triangle, with certain conditions applied:
- P and R are placed such that the line joining them is parallel to the line joining Q and S.
- P is placed on the side opposite to the corner T.
- S and U cannot be placed on the same side.
Step 2: Analyzing the positions.
- The condition that \( \overline{PR} \parallel \overline{QS} \) implies a specific geometric relation between these points.
- P is placed on the side opposite T, which rules out P being at a corner.
- The condition that S and U cannot be placed on the same side implies that at least one of them must be placed at a corner.
Step 3: Conclusion.
Since S and U cannot be on the same side, and S must be placed at a corner (because it cannot be placed at the mid-point), it follows that S cannot be placed at a corner. Therefore, the correct answer is (B).