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:

The point equidistant from all three sides of a triangle is called the incenter. This point is the center of the triangle's inscribed circle, and it is located at the intersection of the triangle's angle bisectors. Option A incorrectly combines angle bisectors and perpendicular bisectors (the latter find the circumcenter, equidistant from vertices). Option C refers to the circumcenter, which is equidistant from the triangle's vertices, not sides. Option D does not guarantee equidistance to all three sides. Option E refers to the centroid, which is the intersection of medians and relates to triangle balance, not side equidistance.

Answer:

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