Question

question 9 of 10 in the diagram below, (overline{rs}) is the perpendicular bisector of (overline{pq}). which of the following statements must be true? check all that apply. a. (rt = st) b. (t) is the mid - point of (overline{pq}) c. (pt=qt) d. (overline{pq}) is the perpendicular bisector of (overline{rs}) e. (mangle ptr = 90^{circ})

Explanation:

Step1: Recall definition of perpendicular bisector

A perpendicular bisector of a line - segment divides the line - segment into two equal parts and is perpendicular to it.

Step2: Analyze option A

There is no information given to suggest that \(RT = ST\). Just because \(\overline{RS}\) is the perpendicular bisector of \(\overline{PQ}\), we cannot conclude this.

Step3: Analyze option B

Since \(\overline{RS}\) is the perpendicular bisector of \(\overline{PQ}\), by definition, the point of intersection \(T\) is the mid - point of \(\overline{PQ}\). So \(T\) is the midpoint of \(\overline{PQ}\), and \(PT=QT\).

Step4: Analyze option C

As \(T\) is the mid - point of \(\overline{PQ}\) (from the property of the perpendicular bisector), \(PT = QT\).

Step5: Analyze option D

There is no information to suggest that \(\overline{PQ}\) is the perpendicular bisector of \(\overline{RS}\).

Step6: Analyze option E

Since \(\overline{RS}\) is the perpendicular bisector of \(\overline{PQ}\), \(\angle PTR = 90^{\circ}\) because of the perpendicularity property.

Answer:

B. \(T\) is the midpoint of \(\overline{PQ}\)
C. \(PT = QT\)
E. \(m\angle PTR=90^{\circ}\)