GATE CS 2022 Set 1 — Question 9
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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.
Correct answer
(B) S cannot be placed at a corner
Solution
Name the corners and the mid-points . Each side contains three of the six points (two corners and one mid-point).Condition 1 (PR QS). Among these six points, two segments with four distinct endpoints are parallel only when one of them joins two mid-points (a mid-segment) and the other lies along the third side, e.g. , where the second segment uses two of . So one of , is a pair of mid-points and the other is two points of the remaining side.Condition 2. is a corner and lies on the side opposite .Condition 3. No side contains both and .Case 1: are the two mid-points, say , and lie on side . Then is a mid-point.Case 2: are the two mid-points, say , . The corner opposite 's side is . Then are two of with already used, so and . Since and share side , , so , again a mid-point.In every valid arrangement is a mid-point, so S cannot be placed at a corner.Checking the other options:
- (A) False: Case 2 places at a mid-point, but Case 1 allows (a corner, with opposite side ).
- (C) False: in Case 1 we can take , , , which satisfies all conditions with at a mid-point.
- (D) False: in Case 1 with , , is a corner.
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