the city council is planning to construct a park on north street that has a triangular perimeter. they want…

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?\na. at the point of intersection of the angle bisectors and perpendicular bisectors of the park\nb. at the center of the inscribed circle of the park\nc. at the center of the circumscribed circle of the park\nd. at the point of intersection of the lines perpendicular to two sides of the park\ne. at the point of intersection of the medians of the park

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?\na. at the point of intersection of the angle bisectors and perpendicular bisectors of the park\nb. at the center of the inscribed circle of the park\nc. at the center of the circumscribed circle of the park\nd. at the point of intersection of the lines perpendicular to two sides of the park\ne. at the point of intersection of the medians of the park

Answer

Explanation:

Step1: Recall triangle center properties

The point equidistant from all three sides of a triangle is the incenter, which is the intersection of angle bisectors. Wait, no—wait, the set of points equidistant from two sides is the angle bisector, but equidistant from all three sides: the incenter is the center of the inscribed circle (incircle), and the incircle's center (incenter) is at the intersection of angle bisectors? Wait, no, the perpendicular bisectors of sides give circumcenter (equidistant from vertices), angle bisectors give incenter (equidistant from sides). Wait, the question is equidistant from all three sides. So the incenter is the center of the inscribed circle, and it's the intersection of angle bisectors? Wait, no, the incenter is the intersection of angle bisectors, and it's equidistant from all sides. Wait, but option B says "at the center of the inscribed circle of the park"—the center of the inscribed circle (incenter) is equidistant from all sides. Wait, let's re-examine the options:

  • Option A: intersection of angle bisectors and perpendicular bisectors? No, angle bisectors and perpendicular bisectors are different.
  • Option B: center of inscribed circle (incenter) – incenter is equidistant from all sides.
  • Option C: center of circumscribed circle (circumcenter) – equidistant from vertices, not sides.
  • Option D: intersection of perpendiculars to two sides (perpendicular bisectors? No, perpendicular to sides at their midpoints are perpendicular bisectors, which give circumcenter. Wait, no—perpendicular to a side at a point: if we take perpendiculars to two sides (not at midpoints), that's not a standard center. Wait, no, the set of points equidistant from two sides is the angle bisector. Wait, maybe I confused. Let's recall:

In a triangle, the incenter (center of incircle) is the intersection of angle bisectors, and it's equidistant from all three sides. The incircle is tangent to all three sides, so its center is equidistant from each side (the distance is the radius). So the center of the inscribed circle is equidistant from all three sides. So option B is correct? Wait, but let's check the options again.

Wait, the problem says "equidistant from all three sides of the park (triangle)". The center of the inscribed circle (incenter) is exactly that point. So option B: "at the center of the inscribed circle of the park" is correct. Wait, but let's check other options:

  • Option A: intersection of angle bisectors and perpendicular bisectors – that's not a standard center.
  • Option B: center of inscribed circle – incenter, equidistant from sides.
  • Option C: center of circumscribed circle – equidistant from vertices, not sides.
  • Option D: intersection of lines perpendicular to two sides – if we take perpendiculars to two sides (not at midpoints), that's not a standard center. If they are perpendicular bisectors (perpendicular to sides at midpoints), that's circumcenter.
  • Option E: intersection of medians – centroid, which is the center of mass, not equidistant from sides.

So the correct answer is B. Wait, but let's confirm: the incenter is the center of the inscribed circle, and it's equidistant from all three sides. So option B is correct.

Step2: Confirm the property

The center of the inscribed circle (incenter) of a triangle is equidistant from all three sides of the triangle. This is because the inscribed circle (incircle) is tangent to all three sides, so the distance from the incenter to each side is the radius of the incircle, hence equal for all sides. Therefore, the council should place the fountain at the center of the inscribed circle of the park.

Answer:

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