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 { 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 segment bisector

A segment bisector divides a segment into two equal parts. Since \( \overline{OS}\cong\overline{SR}\), \(\overline{ST}\) bisects \(\overline{OR}\), so \(\overline{ST}\) is a segment bisector.

Step2: Analyze the definition of perpendicular bisector

A perpendicular bisector is a segment bisector that is also perpendicular to the segment. As \(\angle OST = 90^{\circ}\), \(\overline{ST}\) is perpendicular to \(\overline{OR}\) and bisects it, so \(\overline{ST}\) is a perpendicular bisector.

Step3: Check the vertex of right - angle

Since \(\angle OST=90^{\circ}\), \(S\) 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.