The corners and mid-points of the sides of a triangle are named using distinct letters P, Q, R, S, T

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GATE-CS is a competitive examination conducted in India. This question is from the 2022 GATE-CS Aptitude paper.

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Question

The corners and mid-points of the sides of a triangle are named using distinct letters P, Q, R, S, T, U. Conditions: (1) Line joining P and R is parallel to line joining Q and S. (2) P is on the side opposite to corner T. (3) S and U cannot be on the same side. Which statement is correct?

Accepted Answer

Correct Answer: (B) S cannot be placed at a cornerConcept -- Geometric Reasoning with Named Points: A triangle has 3 corners and 3 mid-points (6 named positions total). From condition (2), P is on the side opposite to corner T, so P is on a side -- either a mid-point of that side. From condition (1), PR is parallel to QS. If both P and R are mid-points of sides, and PR is a midsegment, then QS must also be parallel to PR. For QS to be parallel to PR, Q and S must be positioned such that the line QS is a midsegment or parallel segment. Analysis of the constraints shows that S must be a mid-point (not a corner) for the parallelism condition to be geometrically satisfiable given all the other constraints. Therefore S cannot be placed at a corner.

The corners and mid-points of the sides of a triangle are named using distinct letters P, Q, R, S, T, U. Conditions: (1) Line joining P and R is parallel to line joining Q and S. (2) P is on the side opposite to corner T. (3) S and U cannot be on the same side. Which statement is correct?

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The corners and mid-points of the sides of a triangle are named using distinct letters P, Q, R, S, T, U. Conditions: (1) Line joining P and R is parallel to line joining Q and S. (2) P is on the side opposite to corner T. (3) S and U cannot be on the same side. Which statement is correct?

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