Point of Concurrency

By: Victor Kalev

Real Life Problem

The Cowboy Construction company is building a shopping center. The architect wants to include an information desk that is easily accessible from each of the three mall entrances. The coordinates of the South entrance are (-3, -4), the West entrance is located at (-5, 2), and the East entrance has coordinates of (5, 2). Where should the architect place the information desk if he wants it to be equidistant from all of the entrances?
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Construction of the Triangle

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  1. Using the given coordinates, the vertices of the triangle were found, labeled, and then connected.
  2. The sharp end of the compass was placed on point S, extended past halfway of segment SE, and an arc was made.
  3. Keeping the same compass length, the sharp end was placed at point E and another arc was made.
  4. A line was drawn through the two intersection points of the arcs.
  5. Steps 2 through 4 were repeated for sides EW and WS.
  6. All three lines intersected at a single point, which is the circumcenter.

Calculations

Calculations of Perpendicular Bisectors for the Three Segments:
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Calculation of Intersections of the Three Perpendicular Bisectors:
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Explanation of Solution

The architect of the mall wanted the information desk to be located at the same distance from each of the mall's three entrances (West, East, and South). The point of concurrency that best fits this scenario is the circumcenter because it is equidistant from the vertices of a triangle, which in this case was formed by the entrances. The circumcenter is also the intersection point of the perpendicular bisectors of a triangle. Therefore, the circumcenter was located using these perpendicular bisectors. Each of the bisectors was determined as described above in the Calculations Section and then the circumcenter was found by finding their point of intersection. Finally, the architect ended up with the perfect location for the mall's information desk!