Question

the city council is planning to construct a park on north street that has a triangular perimeter. they want to place a fountain at a point equidistant from all three sides of the park. where should the council place the fountain?
a. at the point of intersection of the angle bisectors and perpendicular bisectors of the park
b. at the center of the inscribed circle of the park
c. at the center of the circumscribed circle of the park
d. at the point of intersection of the lines perpendicular to two sides of the park
e. at the point of intersection of the medians of the park

Explanation:

To determine the location equidistant from all three sides of a triangle (the park), we use the property of angle bisectors. The incenter, which is the intersection of the angle bisectors of a triangle, is equidistant from all three sides. Let's analyze each option:

• Option A: The intersection of angle bisectors and perpendicular bisectors is not a standard point. Angle bisectors give equidistance to sides, perpendicular bisectors to vertices. So A is incorrect.
• Option B: The center of the inscribed circle (incenter) is the intersection of angle bisectors and is equidistant from all sides. This matches the requirement.
• Option C: The circumscribed circle's center (circumcenter) is equidistant from vertices, not sides. So C is incorrect.
• Option D: Perpendicular to two sides' intersection is the circumcenter (for perpendicular bisectors) or related to right angles, not equidistant to all sides. So D is incorrect.
• Option E: The intersection of medians is the centroid, which is about balancing, not equidistance to sides. So E is incorrect.

Answer:

B. at the center of the inscribed circle of the park