Question

each week, marcia travels to the office where she works, the supermarket, and a local fitness center. the three locations represent the vertices of a triangle. marcia wants to move to an apartment that is equidistant from these three places. where should her new apartment be located?
a. the center of the inscribed circle of the triangle
b. the center of the circumscribed circle of the triangle
c. the point of intersection of the angle bisectors for the triangle
d. the point of intersection of the perpendicular bisectors and the angle bisectors for the triangle

Explanation:

Step1: Recall triangle center definitions

The circumcenter (center of the circumscribed circle) of a triangle is the point equidistant from all three vertices.

Step2: Match to problem requirement

Marcia needs a point equidistant from 3 triangle vertices (her 3 locations), so this is the circumcenter.

Step3: Eliminate other options

• A/Incenter: Equidistant to sides, not vertices.
• C/Angle bisector intersection: This is the incenter, not equidistant to vertices.
• D: Perpendicular bisectors meet at circumcenter; angle bisectors meet at incenter—their intersection only occurs in equilateral triangles, and this is not the general point equidistant to vertices.

Answer:

B. the center of the circumscribed circle of the triangle