Question

look at the diagram.
which term describes \\(\overline{qs}\\)?
perpendicular bisector
angle bisector
altitude
none of these

Explanation:

Step1: Recall Definitions

- **Perpendicular Bisector**: A line that is perpendicular to a segment and bisects it (divides into two equal parts).
- **Altitude**: A perpendicular segment from a vertex to the line containing the opposite side.
- **Angle Bisector**: A line that divides an angle into two equal angles.

Step2: Analyze \(\overline{QS}\)

- \(PQ = QR = 12\), so \(Q\) is the midpoint of \(PR\). \(\overline{QS}\) is perpendicular to \(PR\) (right angle at \(Q\)) and connects \(Q\) (on \(PR\)) to \(S\) (on \(RT\)). Wait, no—wait, the diagram: \(PR\) is a vertical segment with \(PQ = QR = 12\), so \(Q\) is the midpoint. \(\overline{QS}\) is perpendicular to \(PR\) (right angle) and bisects \(PR\) (since \(PQ = QR\)). Wait, but also, in triangle \(PRT\), \(\overline{QS}\) is perpendicular to \(PR\) and bisects it. But wait, the options: let's check again. Wait, the altitude is a perpendicular from a vertex to the opposite side. But \(Q\) is on \(PR\), so \(\overline{QS}\) is perpendicular to \(PR\) and bisects it (since \(PQ = QR\)), so it's a perpendicular bisector? Wait, no—wait, the original diagram: \(PR\) is length \(24\) (12 + 12), \(Q\) is the midpoint, and \(QS\) is perpendicular to \(PR\). So a perpendicular bisector is a line that is perpendicular to a segment and bisects it. So \(\overline{QS}\) is perpendicular to \(PR\) and bisects \(PR\) (since \(PQ = QR\)), so it's a perpendicular bisector? Wait, but the option "perpendicular bisector"—wait, maybe I made a mistake. Wait, let's re-express:

Wait, the problem is about \(\overline{QS}\). Let's check the definitions again:

- **Perpendicular Bisector**: A line that is perpendicular to a segment and passes through its midpoint (bisects it). Since \(PQ = QR = 12\), \(Q\) is the midpoint of \(PR\), and \(\overline{QS}\) is perpendicular to \(PR\) (right angle at \(Q\)), so \(\overline{QS}\) is the perpendicular bisector of \(PR\). Wait, but the options include "perpendicular bisector", "altitude", "angle bisector", "none of these".

Wait, maybe I misread the diagram. Let's see: \(PR\) is a vertical side, \(Q\) is the midpoint, \(QS\) is horizontal (perpendicular to \(PR\)) and goes to \(S\) on \(RT\). So in triangle \(PRT\), \(QS\) is perpendicular to \(PR\) and bisects \(PR\), so it's the perpendicular bisector. But wait, the altitude of a triangle is a perpendicular segment from a vertex to the opposite side. But \(Q\) is on \(PR\), not a vertex (the vertices are \(P\), \(R\), \(T\)). Wait, maybe the triangle is \(PRT\), with vertices \(P\), \(R\), \(T\). Then \(PR\) is a side, and \(Q\) is the midpoint of \(PR\), and \(QS\) is perpendicular to \(PR\) and connects to \(S\) on \(RT\). So \(QS\) is the perpendicular bisector of \(PR\) because it's perpendicular to \(PR\) and bisects it (since \(PQ = QR\)). So the correct term is "perpendicular bisector"? Wait, but the initial thought was altitude, but altitude is from a vertex. Wait, maybe the triangle is \(QRT\) or \(PQT\)? No, the diagram shows \(P\), \(Q\), \(R\) with \(PQ = QR = 12\), right angle at \(Q\) for \(QS\) and \(PR\). So \(QS\) is perpendicular to \(PR\) and bisects it (since \(Q\) is the midpoint), so it's a perpendicular bisector.

Wait, but let's check the options again. The options are:

- perpendicular bisector
- altitude
- angle bisector
- none of these

So let's re-express:

- **Perpendicular Bisector**: Correct, because \(QS\) is perpendicular to \(PR\) and bisects \(PR\) (since \(PQ = QR\)).
- **Altitude**: An altitude is from a vertex, but \(Q\) is not a vertex of the main triangle (if the triangle is \(PRT\), vertices are \(P\), \(R\), \(T\); \(Q\) is on \(PR\)), so \(QS\) is not an altitude (altitude would be from \(P\) or \(R\) or \(T\) to the opposite side).
- **Angle Bisector**: \(QS\) does not divide an angle into two equal parts (no angle is shown being bisected here).

So the correct term is "perpendicular bisector". Wait, but maybe I made a mistake. Wait, the original diagram: \(PR\) is a vertical segment, \(Q\) is the midpoint, \(QS\) is horizontal (perpendicular) to \(PR\), so \(QS\) is the perpendicular bisector of \(PR\). So the answer should be "perpendicular bisector".

Wait, but the initial thought was altitude, but altitude is from a vertex. Since \(Q\) is on \(PR\), not a vertex, \(QS\) is not an altitude. So the correct option is "perpendicular bisector".

Answer:

perpendicular bisector