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?

Question

Prepp Team · Accepted Answer

Let the triangle corners be A, B, C and the mid-points of the opposite sides be D, E, F respectively. We are assigning 6 distinct letters (P, Q, R, S, T, U) to these 6 positions.

Analyzing the Conditions

Condition 1: The line joining P and R is parallel to the line joining Q and S (PR || QS).
Condition 2: P is placed on the side opposite to the corner T.
Condition 3: S and U cannot be placed on the same side.

Deduction from Condition 1

The condition PR || QS strongly suggests a relationship based on the Midpoint Theorem. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side. For instance, DE || AB, EF || BC, FD || AC.

If PR || QS, a likely scenario is that P and R represent two corners (e.g., A and B) and Q and S represent the midpoints of the other two sides (e.g., D and E, the midpoints of BC and AC respectively). In this case, the line segment DE is parallel to the side AB (which connects the two corners P and R).

Therefore, we can hypothesize:

{P, R} = {Corners A, B}
{Q, S} = {Midpoints D, E} (midpoints of sides opposite A and B)

Deduction from Condition 2

P is on the side opposite to corner T.

Based on the hypothesis from Condition 1, P is a corner (A or B). The side opposite a corner is the side connecting the other two corners. If P=A and R=B, the third corner is C. The side opposite C is AB.

Since P (A or B) is one of the endpoints of the side AB, P lies on the side opposite corner C. This implies that T must be the third corner (C).

So, T = C.

Determining Positions of Remaining Letters

Letters used so far: P, R (corners A, B), T (corner C), Q, S (midpoints D, E).

The only remaining letter is U. The only remaining position is the midpoint of the side opposite T (corner C), which is the midpoint of AB. Let this be F.

Therefore, U = F (midpoint of AB).

Checking Condition 3

S and U cannot be placed on the same side.

We have U = F (midpoint of AB). S is either D (midpoint of BC) or E (midpoint of AC).

If S=D, U=F. S is on side BC, U is on side AB. They are not on the same side segment.
If S=E, U=F. S is on side AC, U is on side AB. They are not on the same side segment.

Condition 3 holds true in this configuration.

Summary of a Valid Configuration

A valid assignment satisfying all conditions is:

Corners: {P, R, T} = {A, B, C}
Midpoints: {Q, S, U} = {D, E, F} (where {Q, S} are midpoints of sides adjacent to T, and U is the midpoint of the side opposite T)

In this configuration, S is assigned to a midpoint (either D or E).

Evaluating the Options

P cannot be placed at a corner: This is false, as our valid configuration places P at a corner.
S cannot be placed at a corner: This is true, as our valid configuration places S at a midpoint.
U cannot be placed at a mid-point: This is false, as our valid configuration places U at a midpoint (F).
R cannot be placed at a corner: This is false, as our valid configuration places R at a corner.

Based on the consistent deduction, S must be a midpoint.