The corners and mid-points of the sides of a triangle are named using the…
2022
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?
- A.
P cannot be placed at a corner
- B.
S cannot be placed at a corner
- C.
U cannot be placed at a mid-point
- D.
R cannot be placed at a corner
Show answer & explanation
Correct answer: B
Answer: S cannot be placed at a corner.
Interpretation: The six labels P, Q, R, S, T, U are used for the three corners and the three midpoints of the sides.
Use the parallelism: If the segment joining P and R is parallel to the segment joining Q and S, one consistent geometric situation is that P and R are the two endpoints of one side (i.e. two corners), while Q and S are the midpoints of the other two sides. In a triangle, the segment joining the midpoints of two sides is parallel to the third side.
Consequence: In that arrangement Q and S must be midpoints, so S cannot be a corner.
Check compatibility with the other conditions: "P is placed on the side opposite to the corner T" is satisfied by putting P on that side (either as the corner at one end of the side or as its midpoint). "S and U cannot be placed on the same side" can be satisfied by placing U on a different side than S. Thus the arrangement is consistent and forces S to be a midpoint (not a corner).
Therefore the only statement that must be true given all information is that S cannot be placed at a corner.
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