GATE CS 2022 Q9 — General Aptitude

In a triangle, the only parallel lines formed by the corners and midpoints are a side of the triangle (joining two corners) and the line joining the midpoints of the other two sides.

Therefore, one of the pairs $\{P, R\}$ or $\{Q, S\}$ must consist of two corners, and the other must consist of two midpoints.

• Case 1: Suppose $\{Q, S\}$ are corners and $\{P, R\}$ are midpoints.
  • Since $T$ is a corner (as $P$ is opposite to the "corner $T$"), the three corners of the triangle must be $Q$, $S$, and $T$.
  • The side opposite to corner $T$ is the side $QS$.
  • According to the second statement, $P$ must lie on the side opposite to $T$ (i.e., on $QS$). However, the only points on side $QS$ are $Q$, $S$, and their midpoint $U$. Since $P$ is one of the other midpoints, $P$ cannot lie on $QS$, leading to a contradiction.
• Case 2: Suppose $\{P, R\}$ are corners and $\{Q, S\}$ are midpoints.
  • This configuration is consistent and avoids any contradiction.
  • Since $\{Q, S\}$ are midpoints, $S$ must be a midpoint.

Thus, $S$ cannot be placed at a corner.