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: 1. The line joining P and R is parallel to the line joining Q and S. 2. P is placed on the side opposite to the corner Q. 3. S and U cannot be placed on the same side. Which one of the following statements about the above information is correct? 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

GATE 2022 · General Aptitude · Triangles · medium

Answer: S cannot be placed at a corner (answer B)

1. Interpret Statement 2: A point on a side (not at a vertex) is a mid-point in this setup. So P must be a mid-point, and Q must be a corner.
2. Interpret Statement 1 with P as mid-point: Since Q is a corner and P is a mid-point (opposite Q's side), for PR to be parallel to QS, the pair (Q,S) must mirror (P,R). Since Q is a corner, S must be the mid-point of the side that creates the parallel: S must be a mid-point.
3. Confirm S cannot be at a corner: From Statement 1, S must be a mid-point (not a corner) to maintain the parallel condition with Q (corner). Checking statement 3 (S and U on different sides) is consistent with S being a mid-point. Thus S cannot be placed at a corner.