Question

consider the diagram. which line segment has the same measure as \\( \overline{tq} \\)? \\( \overline{sr} \\) \\( \overline{tr} \\) \\( \overline{st} \\) \\( \overline{qs} \\)

Explanation:

Step1: Recall the property of perpendicular bisector

A perpendicular bisector of a line segment divides the line segment into two equal - length parts. In the given diagram, line \(m\) is the perpendicular bisector of \(QR\).

Step2: Apply the property of congruent triangles

We can consider triangles \(SQR\) and \(TQR\). Since \(m\) is the perpendicular bisector of \(QR\), we know that \(QS = RS\) and \(QT=RT\) (by the property of the perpendicular bisector: any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment).

Answer:

\(\overline{TR}\)