Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)

answer
$overline{rs}$ is a perpendicular bisector.
$r$ is the vertex of a right angle.
$s$ is the vertex of a right angle.
$r$ is the midpoint of a segment in the diagram.
$s$ is the midpoint of a segment in the diagram.
none of the above.

Explanation:

1. $\overline{RS}$ is a perpendicular bisector: A perpendicular bisector must be perpendicular to a segment and pass through its midpoint. While $\overline{RS}$ is perpendicular to $\overline{PQ}$ at $R$, we only know $R$ divides $\overline{PQ}$ into two congruent segments (so $R$ is the midpoint of $\overline{PQ}$), but there is no indication $\overline{RS}$ bisects another segment, so this is not true.
2. $R$ is the vertex of a right angle: The right angle symbol at $R$ confirms $\overline{RS} \perp \overline{PQ}$, so $\angle PRS$ (or $\angle QRS$) is a right angle with vertex $R$. This is true.
3. $S$ is the vertex of a right angle: There is no right angle symbol or evidence of perpendicular lines meeting at $S$, so this is not true.
4. $R$ is the midpoint of a segment in the diagram: The two congruent tick marks on $\overline{PQ}$ show $PR = RQ$, so $R$ is the midpoint of $\overline{PQ}$. This is true.
5. $S$ is the midpoint of a segment in the diagram: There are no congruent tick marks or other indicators that $S$ divides any segment into two equal parts, so this is not true.

Answer:

R is the vertex of a right angle.
R is the midpoint of a segment in the diagram.