Let's analyze the statements given in each of the options and determine which one aligns correctly with the problem.
To prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side, we should focus on the properties of midpoints and parallel lines in triangles.
Given a triangle [tex]\( \triangle PQR \)[/tex]:
1. Let's denote [tex]\( S \)[/tex] and [tex]\( T \)[/tex] as the midpoints of sides [tex]\( \overline{PQ} \)[/tex] and [tex]\( \overline{PR} \)[/tex], respectively.
2. According to the midpoint theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.
The appropriate statement would be:
- If [tex]\( S \)[/tex] and [tex]\( T \)[/tex] are midpoints, which means [tex]\( PS = SQ \)[/tex] and [tex]\( PT = TR \)[/tex], then segment [tex]\( \overline{ST} \)[/tex] is parallel to [tex]\( \overline{QR} \)[/tex].
Let's examine each option:
(A) In [tex]\( \triangle PQR \)[/tex], if [tex]\( \overline{ST} \parallel \overline{QR} \)[/tex], then [tex]\( PS = SQ \)[/tex] and [tex]\( PT = TR \)[/tex].
- This suggests that parallelism leads to midpoints' property, which is incorrect.
(B) In [tex]\( \triangle PQR \)[/tex], if [tex]\( PS = SQ \)[/tex] and [tex]\( PT = TR \)[/tex], then [tex]\( \overline{ST} \parallel \overline{QR} \)[/tex].
- This correctly aligns with the midpoint theorem, stating that if midpoints are given, then the segment joining them is parallel to the third side. Hence, this is correct.
(C) In [tex]\( \triangle PQR \)[/tex], if [tex]\( \overline{ST} \parallel \overline{QR} \)[/tex], then [tex]\( PS = PT \)[/tex] and [tex]\( PQ = PR \)[/tex].
- This suggests an incorrect relationship as it implies equal segments in other pairs, which is not the requirement for the segment joining two midpoints.
(D) In [tex]\( \triangle PQR \)[/tex], if [tex]\( PS = PT \)[/tex] and [tex]\( PQ = PR \)[/tex], then [tex]\( \overline{ST} \parallel \overline{QR} \)[/tex].
- This does not correctly state the conditions of the midpoints property. It suggests equal lengths which are not necessarily related to the property of midpoints and the parallel segment.
Thus, the correct rephrased statement for Wei's proof is:
(B) In [tex]\( \triangle PQR \)[/tex], if [tex]\( PS = SQ \)[/tex] and [tex]\( PT = TR \)[/tex], then [tex]\( \overline{ST} \parallel \overline{QR} \)[/tex].
