Question

question 12 of 45
\\(\overline{tz}\\) is a midsegment, which of the following statements cannot be true?
a. \\(\overline{qt}<\overline{tr}\\)
b. \\(\overline{ux}\\) is a perpendicular bisector
c. \\(\overline{rs}\\) is twice as long as \\(\overline{tz}\\)
d. \\(z\\) is a mid - point

Explanation:

Step1: Recall mid - segment properties

A mid - segment of a triangle is parallel to the third side and its length is half of the length of the third side. In \(\triangle RQS\), if \(\overline{TZ}\) is a mid - segment, then \(\overline{TZ}\parallel\overline{RS}\) and \(RS = 2TZ\). Also, the endpoints of the mid - segment are mid - points of the two sides of the triangle.

Step2: Analyze option A

\(\overline{QT}<\overline{TR}\) is possible. The lengths of the segments on the sides of the triangle can vary in such a way that this inequality holds.

Step3: Analyze option B

A mid - segment has nothing to do with being a perpendicular bisector. A mid - segment is a line segment connecting the mid - points of two sides of a triangle and is parallel to the third side. It does not bisect any side at a right angle in general.

Step4: Analyze option C

Since \(\overline{TZ}\) is a mid - segment, by the mid - segment theorem, \(\overline{RS}\) is twice as long as \(\overline{TZ}\), so this statement is true.

Step5: Analyze option D

Since \(\overline{TZ}\) is a mid - segment, the endpoints \(T\) and \(Z\) are mid - points of the sides of the triangle, so \(Z\) is a mid - point. This statement is true.

Answer:

B. \(\overline{UX}\) is a perpendicular bisector