Question

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

Explanation:

Step1: Recall definition of perpendicular bisector

A perpendicular bisector of a line - segment is perpendicular to the line - segment and divides it into two equal parts.
Since $\overline{RS}$ is the perpendicular bisector of $\overline{PQ}$, by definition, $\overline{RS}\perp\overline{PQ}$ and $T$ is the mid - point of $\overline{PQ}$, so $PT = QT$. Also, since $\overline{RS}\perp\overline{PQ}$, $\angle PTR=90^{\circ}$.
There is no information given to suggest that $RT = ST$ or that $\overline{PQ}$ is the perpendicular bisector of $\overline{RS}$.

Step2: Analyze each statement

• $PT = QT$: True because $T$ is the mid - point of $\overline{PQ}$ (by the definition of a bisector).
• $\overline{RS}\perp\overline{PQ}$: True by the definition of a perpendicular bisector.
• $T$ is the mid - point of $\overline{PQ}$: True by the definition of a bisector.
• $RT = ST$: No information to support this.
• $\overline{PQ}$ is the perpendicular bisector of $\overline{RS}$: No information to support this.
• $m\angle PTR = 90^{\circ}$: True because $\overline{RS}\perp\overline{PQ}$.

Answer:

PT = QT, RS ⊥ PQ, T is the midpoint of PQ, m∠PTR = 90°