Triangle town wants to build a school that is equidistant from its three cities L, M, and N. Which construction correctly finds the best location of the school? (1 point)

Question

Construction X is triangle LMN with an inscribed circle. Point A is on segment LM, point D is on segment MN, and point B is on segment LN, segment AN bisects angle N, segment MN bisects angle M, the angle formed by MDL is a right angle, and point C is the intersection of segments MB, LD, and AN.
Construction X
Construction Y is triangle LMN within a circumscribed circle. Point A is on segment LM, point B is on segment MN, point D is on segment MB, point C is on segment LN, angle MAB is a right angle, angle NCD is a right angle, and point E is the intersection of segments AB and EC.
Construction Y

Construction Y because point E is the incenter of ΔLMN

Construction Y because point E is the circumcenter of ΔLMN

Construction X because point C is the incenter of ΔLMN

Construction X because point C is the circumcenter of ΔLMN

Answer 1

To determine the best location for the school that is equidistant from the three cities L, M, and N, we need to understand the concepts of the incenter and circumcenter of a triangle.

1. Incenter: The incenter is the point where the angle bisectors of a triangle intersect, and it is equidistant from all three sides of the triangle. This is not necessarily equidistant from the vertices of the triangle.

2. Circumcenter: The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is equidistant from all three vertices of the triangle.

Since we want a location that is equidistant from the three cities (the vertices of the triangle), we are looking for the circumcenter, not the incenter.

Looking at the provided constructions:

• Construction X describes the incenter.
• Construction Y describes the circumcenter.

Based on this analysis:

Construction Y because point E is the circumcenter of ΔLMN is the correct answer.