Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)
answer
$overline{qr}$ is a segment bisector.
$overline{qr}$ is a perpendicular bisector.
$overline{qr}$ is an angle bisector.
$q$ is the vertex of a right angle.
$r$ is the vertex of a right angle.
$r$ is the midpoint of a segment in the diagram.

Explanation:

1. $\overline{QR}$ is a segment bisector: The diagram shows $PQ = QM$, so $Q$ bisects $\overline{PM}$, making $\overline{QR}$ a segment bisector of $\overline{PM}$.
2. $\overline{QR}$ is a perpendicular bisector: A perpendicular bisector must be perpendicular to the segment. $\overline{RP} \perp \overline{PM}$, but $\overline{QR}$ is not perpendicular to $\overline{PM}$, so this is false.
3. $\overline{QR}$ is an angle bisector: There is no indication $\overline{QR}$ splits any angle into two equal parts, so this is false.
4. $Q$ is the vertex of a right angle: $\angle PQR$ is not a right angle (as $\overline{QR}$ is not perpendicular to $\overline{PM}$), so this is false.
5. $R$ is the vertex of a right angle: The right angle symbol at $\angle RPO$ confirms $\angle PRP$ (actually $\angle OPR$) is right, but $R$ is the vertex of $\angle ORP$, which is not marked right. However, $\angle PRQ$ is not right, but the diagram shows $\overline{RP} \perp \overline{PM}$, so $R$ is part of that right angle, but the statement is about $R$ being the vertex. Wait, no: the right angle is at $P$, so $P$ is the vertex. So $R$ is not the vertex of a right angle. Wait, no: the right angle is at $P$, so $P$ is the vertex. So $R$ is not the vertex of a right angle. So this is false.
6. $R$ is the midpoint of a segment in the diagram: There is no indication $R$ divides any segment into two equal parts, so this is false.

Answer:

$\overline{QR}$ is a segment bisector.