Question

PLSS HELP !! 100 points 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?
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

The construction correctly finds the best location of the warehouse is shown in option C (Construction Y because point E is the circumcentre of triangle LMN and due to its exact equidistant location from the three stores at L, M, and N, Point E is the ideal location for the warehouse.)
What is circumcentre?
The location that is equally spaced from the three vertices and where the perpendicular bisectors of a triangle's sides intersect.
Calculation for distance of the three stores from warehouse:
Precisely one circle crosses through all three, which are all on the same line. Finding the centre of the circle that goes through all three of the points is equivalent to finding the point that is equally distant from each of the three points (since all points on a circle are equidistant from the centre).
L, M, and N are our points. To create the triangle, draw the lines LM, LN, and MN. The junction of any two of the lines' perpendicular bisectors, point E, will serve as the circle's centre.
Due to the fact that E is the triangle's LMN's circumcenter, as demonstrated in Construction Y.
Therefore, given that it is precisely equidistant between the three stores at L, M, and N, this is the optimal location for the warehouse