To solve the problem, we need to understand the geometric relationship between the points P, Q, and X. The statement "XP = XQ" means that the distance from point X to point P is equal to the distance from point X to point Q. This condition describes a specific set of points in the plane. ### Step-by-Step Explanation 1. **Understanding the Condition**: - The condition \( XP = XQ \) indicates that point X is equidistant from points P and Q. This means that for any position of X, the distance to P is the same as the distance to Q. 2. **Geometric Interpretation**: - The set of all points that are equidistant from two fixed points (in this case, P and Q) forms a specific geometric figure. This figure is known as the **perpendicular bisector** of the line segment connecting P and Q. 3. **Why the Perpendicular Bisector?**: - To find the perpendicular bisector, we first identify the midpoint M of the segment PQ. The perpendicular bisector is a line that: - Passes through this midpoint M. - Is perpendicular to the line segment PQ. - Any point on this line (the perpendicular bisector) will have the same distance to P and Q, satisfying the condition \( XP = XQ \). 4. **Mathematical Representation**: - If we denote the coordinates of P as \( (x_1, y_1) \) and Q as \( (x_2, y_2) \), the midpoint M can be calculated as: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] - The slope of line PQ is given by: \[ \text{slope of PQ} = \frac{y_2 - y_1}{x_2 - x_1} \] - The slope of the perpendicular bisector will be the negative reciprocal of this slope. 5. **Conclusion**: - Therefore, the locus of point X, which satisfies the condition \( XP = XQ \), is indeed the perpendicular bisector of the segment connecting points P and Q. ### Evaluating Other Options - **Option A: A straight line**: - This is too vague. While the perpendicular bisector is a straight line, not all straight lines would satisfy the condition \( XP = XQ \). Only the specific line that is the perpendicular bisector does. - **Option B: A circle**: - A circle would imply that all points on the circle are equidistant from a single point (the center). In this case, we have two fixed points (P and Q), not one, so this option is incorrect. - **Option C: The bisector of angle PXQ**: - The angle bisector of angle PXQ would not necessarily maintain the condition \( XP = XQ \) for all points X. The angle bisector relates to angles formed by lines from P and Q to X, not distances. Thus, this option is also incorrect. ### Revision Summary - The locus of points X such that \( XP = XQ \) is the **perpendicular bisector** of the segment connecting points P and Q. - The perpendicular bisector is the line that is equidistant from both points P and Q. - Other options do not satisfy the condition of equidistance or are too general. - Understanding the geometric properties of points and lines is crucial in solving such problems.