GATE CS 2022 Set 1 — Question 9
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General Aptitude → Analytical Aptitude → Deduction & Induction
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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
Let's analyze the geometry and statements:
1."The line joining P and R is parallel to the line joining Q and S."
In a triangle, a line segment is parallel to another only if one connects two midpoints and the other connects the two corners of the third side (Midpoint Theorem). Thus, one pair (e.g., P, R) are corners and the other pair (Q, S) are midpoints, or vice versa.2."P is placed on the side opposite to the corner T."
- If P were a midpoint, for PR to be parallel to QS, P and R would likely be midpoints of the sides adjacent to a vertex, and QS would be the base. However, if P is on the side opposite to T, and T is a corner, P lies on the base. This suggests P is a corner on the base line.
- Let's assume P and R are the corners of the base. Then the line PR is the base side.
- For QS to be parallel to PR, Q and S must be the midpoints of the other two sides.
- In this configuration: P and R are corners. Q and S are midpoints.
- T must be the third corner (opposite to side PR). This satisfies "P is placed on the side opposite to the corner T" (since P is an endpoint of that side).
- We have corners: P, R, T. Midpoints: Q, S, U.
- Q and S are midpoints of sides TP and TR (or vice versa).
- U is the midpoint of the base PR.
- S is on side TR (or TP). U is on side PR. They are on different sides. This satisfies the condition.
- Suppose Q and S are corners (base QS) and P, R are midpoints.
- Then P is a midpoint on a side adjacent to the top vertex.
- Statement 2 says P is on the side opposite to corner T. If P is a midpoint, T must be the opposite vertex. But in this setup, the vertex opposite to P's side is one of the base corners (Q or S). So T would be Q or S.
- Statement 3 says S and U cannot be on the same side. If T=S, then S is a corner. U would be the midpoint of the base QS. Thus S and U would both be on the side QS. This contradicts Statement 3.
- The valid configuration is: P and R are corners; Q and S are midpoints.
- Since S is a midpoint, it cannot be placed at a corner.
- Therefore, statement (B) is correct.
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