Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)
answer attempt 2 out of 2
$overline{rs}$ 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. We only know $\overline{RS}$ is perpendicular to $\overline{PQ}$ at $R$, and $R$ is the midpoint of $\overline{PQ}$, but we have no information that $S$ is the midpoint of $\overline{NO}$, so this is not necessarily true.
2. $R$ is the vertex of a right angle: The diagram shows a right angle symbol at $R$, meaning $\overline{RS} \perp \overline{PQ}$, so $R$ is the vertex of a right angle. This is true.
3. $S$ is the vertex of a right angle: There is no right angle symbol or indication that $\overline{RS}$ is perpendicular to $\overline{NO}$ at $S$, so this is not necessarily true.
4. $R$ is the midpoint of a segment in the diagram: The tick marks on $\overline{PR}$ and $\overline{RQ}$ show they are congruent, 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 tick marks or other indications that $S$ divides $\overline{NO}$ into two congruent parts, so this is not necessarily true.

Answer:

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