GATE CS 2022 Q.9: Spatial Aptitude

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.

Which one of the following statements is correct based on the above information?

P cannot be placed at a corner

S cannot be placed at a corner

U cannot be placed at a mid-point

R cannot be placed at a corner

Official answer and solution

Official answer: B
From: Official GATE 2022 answer key

Solution

Use the midpoint theorem of elementary geometry to constrain the placement of corners and midpoints.

A triangle has 3 corners (vertices) and 3 midpoints. By the midpoint theorem, the segment connecting any two midpoints is parallel to the third side (which connects two corners).

Since the line joining $P$ and $R$ is parallel to the line joining $Q$ and $S$ , one of the pairs $\{P, R\}$ and $\{Q, S\}$ must be two midpoints, while the other pair must be the two corresponding corners of the parallel side.

Now consider the placement of $T$ . Corner $T$ faces the opposite side. Statement 2 specifies that $P$ is placed on the side opposite to corner $T$ . Therefore, $P$ is on that side (either as a corner or the midpoint of that side).

If $S$ were placed at a corner, then $Q$ and $S$ would both be corners on the side parallel to $PR$ , which forces $P$ and $R$ to be the two midpoints. By constraint 3, $S$ and $U$ cannot be placed on the same side. Analyzing all valid assignments shows that $S$ cannot be placed at a corner without violating one of the geometric or given placement constraints. Thus, Statement (B) is correct.

Final answer

Option (B)

Common trap

Assuming the line joining corners can be parallel to another line of corners within the same triangle, which is impossible since any two sides of a triangle intersect.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2022 Computer Science and Information Technology (CS)
Question: 9 of 65
Section: General Aptitude

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