Question

question 1 of 10
in the diagram below, $overleftrightarrow{rs}$ is the perpendicular bisector of $overline{pq}$. which of the following statements must be true?
check all that apply.
a. $overline{pq}$ is the perpendicular bisector of $overline{rs}$
b. $mangle ptr = 90^{circ}$
c. $pt = qt$
d. $t$ is the mid - point of $overline{pq}$

Explanation:

Step1: Recall perpendicular - bisector definition

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

Step2: Analyze option A

Just because $\overleftrightarrow{RS}$ is the perpendicular bisector of $\overline{PQ}$, there is no information to suggest that $\overline{PQ}$ is the perpendicular bisector of $\overleftrightarrow{RS}$. So option A is not necessarily true.

Step3: Analyze option B

Since $\overleftrightarrow{RS}$ is the perpendicular bisector of $\overline{PQ}$, by the definition of perpendicular lines, $\angle PTR = 90^{\circ}$, so $m\angle PTR=90^{\circ}$. Option B is true.

Step4: Analyze option C

As $\overleftrightarrow{RS}$ is the perpendicular bisector of $\overline{PQ}$, it divides $\overline{PQ}$ into two equal - length segments. So $PT = QT$. Option C is true.

Step5: Analyze option D

A bisector of a line segment divides the segment into two equal parts, and the point of division is the mid - point. Since $\overleftrightarrow{RS}$ bisects $\overline{PQ}$, $T$ is the mid - point of $\overline{PQ}$. Option D is true.

Step6: Analyze option E

There is no information given to suggest that $PT = ST$. Just because $\overleftrightarrow{RS}$ is the perpendicular bisector of $\overline{PQ}$ does not imply this equality. So option E is not necessarily true.

Answer:

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