Apex Grocery wants to build a warehouse that is equidistant from its three stores, L, M, and N. Which construction correctly finds the best location of the warehouse?

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 X, because point C is the incenter of ΔLMN
Construction Y, because point E is the circumcenter of ΔLMN
Construction X, because point C is the circumcenter of the ΔLMN

Answer 1

To find the best location for the warehouse that is equidistant from stores L, M, and N, we need to determine the type of center that provides this condition.

• The incenter is the point where the angle bisectors of a triangle intersect, and it is the center of the inscribed circle. However, it is not necessarily equidistant from the vertices of the triangle.

• The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is the center of the circumscribed circle. This point is equidistant from all three vertices of the triangle.

In the given constructions:

Construction X describes an inscribed circle and indicates that point C is formed by bisecting angles and is likely connected to the incenter.

Construction Y describes a circumscribed circle and indicates that point E is formed with right angles and intersects line segments in a manner that suggests circumcenter properties.

Since we want the point that is equidistant from the three stores (i.e., the circumcenter), this would be represented in Construction Y.

Therefore, the appropriate statement is:

Construction Y, because point E is the circumcenter of ΔLMN.