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 { s t } \\) is a segment bisector.
\\( \overline { s t } \\) is a perpendicular bisector.
\\( s \\) is the vertex of a right angle.
\\( t \\) is the vertex of a right angle.
\\( t \\) is the midpoint of a segment in the diagram.
none of the above.

Explanation:

Step1: Analyze the definition of a segment bisector

A segment bisector is a line, ray, or segment that divides another segment into two equal parts. Since \( \overline{RS}\cong\overline{SO}\) (marked as equal in the diagram), and \( \overline{ST}\) intersects \( \overline{RO}\), \( \overline{ST}\) is a segment bisector.

Step2: Analyze the definition of a perpendicular bisector

A perpendicular bisector is a segment bisector that is also perpendicular to the segment it bisects. We know \( \overline{ST}\perp\overline{RO}\) (right - angle symbol at \(S\)) and \( \overline{ST}\) bisects \( \overline{RO}\), so \( \overline{ST}\) is a perpendicular bisector.

Step3: Analyze the vertex of a right angle

A right - angle vertex is the point where the two sides of a right angle meet. Since there is a right - angle symbol at \(S\) (\( \angle TSR=\angle TSO = 90^{\circ}\)), \(S\) is the vertex of a right angle.

Answer:

\(\overline{ST}\) is a segment bisector, \(\overline{ST}\) is a perpendicular bisector, \(S\) is the vertex of a right angle.