Final answer:
The epicenter is located at the intersection of circles centered at the receiving stations with radii equal to the given distances.
Step-by-step explanation:
The student's question involves locating the epicenter of an earthquake given the distances from three receiving stations on a coordinate plane. To solve this, we'll use the concept of circles, where each circle represents the set of all points that are a certain distance (the radius) from a single point (the center).
For Station X located at (5,2), we draw a circle with a radius of 5 units. For Station Y at (-5,-9), we draw a circle with a radius of x units. And for Station Z at (-5,7), we have a circle with a radius of 10 units. The epicenter will be a point that lies simultaneously on all three circles, which geometrically is their intersection point.
We do not have enough information to determine x, the distance from Station Y to the epicenter. However, by geometric reasoning, the epicenter must lie to the northwest of Station X because Stations Y and Z are both to the west, and the epicenter must be farther from Y than it is from X.
A more precise location requires further data or calculations, typically involving the use of the distance formula and systems of equations to solve for the point of intersection of the three circles.