### Correct Option: A. the perpendicular bisector of ST #### Detailed Explanation: To understand why the correct answer is option A, we need to delve into the concept of loci and the geometric properties of points. 1. **Definition of Locus**: A locus is a set of points that satisfy a particular condition. In this case, we are looking for the locus of points that are equidistant from two fixed points, S and T. 2. **Equidistant Condition**: For a point P to be equidistant from points S and T, the distance from P to S must equal the distance from P to T. Mathematically, this can be expressed as: \[ PS = PT \] where PS is the distance from point P to point S, and PT is the distance from point P to point T. 3. **Geometric Interpretation**: The set of all points P that satisfy the condition \( PS = PT \) forms a specific geometric figure. This figure is the perpendicular bisector of the line segment ST. 4. **Why the Perpendicular Bisector?**: - The perpendicular bisector of a line segment is defined as the line that is perpendicular to the segment at its midpoint. - Any point on this line is equidistant from the endpoints of the segment (in this case, points S and T). - This is because if you take any point on the perpendicular bisector, the right triangles formed by the midpoint and the endpoints will have equal lengths for the segments connecting the midpoint to S and T, thus satisfying the condition \( PS = PT \). 5. **Visualizing the Perpendicular Bisector**: - Imagine a line segment connecting points S and T. The midpoint of this segment is where the perpendicular bisector intersects it at a right angle (90 degrees). - If you draw a line through this midpoint that is perpendicular to ST, every point on this line will be the same distance from S and T. #### Why the Other Options are Incorrect: - **Option B: the angle bisector of PS and ST**: - The angle bisector divides an angle into two equal angles. However, it does not guarantee that the distances from point P to points S and T are equal. Therefore, this option does not satisfy the equidistant condition. - **Option C: a perpendicular to ST**: - While a perpendicular line can be drawn to ST, it does not necessarily pass through the midpoint of ST. A perpendicular line can be anywhere along the length of ST, and points on this line are not guaranteed to be equidistant from S and T. - **Option D: a line parallel to ST**: - A line parallel to ST will maintain a constant distance from ST but will not ensure that points on this line are equidistant from S and T. The distances will vary depending on the position of the line relative to S and T. ### Summary of Key Points: - The locus of points equidistant from two points S and T is the **perpendicular bisector** of the segment ST. - The perpendicular bisector is the only line that guarantees equal distances from both endpoints. - Other options (angle bisector, arbitrary perpendiculars, and parallel lines) do not satisfy the equidistant condition. - Understanding the properties of geometric figures is crucial for solving locus-related problems in mathematics.