In order to solve the problem presented on the previous page, it is necessary to first develop a problem solving strategy in which we identify the information we need to extrapolate, the technology we will use, and the process by which we will arrive at the solution. First, it will be necessary to understand the geometric concepts of triangle centers and distance presented in the problem. Next, GeoGebra can be used to construct the triangle centers and measure the lengths between the vertices of the triangle (cities) and the centers (stadium). Calculations can then be made to determine the cost of paving a given route from city to stadium, and these calculations will most effectively be organized in a spreadsheet. Finally, we will analyze our spreadsheet to determine the most cost-effective location for the new stadium.
Step 1: Understanding Triangle Centers and Distance between Points
In order to solve the problem, it is crucial to understand how a triangle’s centroid, circumcenter, incenter, and orthocenter are constructed. This information is available in a wealth of places, but this website offers clear explanations and graphics that aid in understanding triangle centers. Using GeoGebra, we can create constructions of the centers that can be manipulated in order to better understand their properties:
The centroid is formed by the intersection of the three line segments that each extend from one vertex to the midpoint of the opposite side (Dunbar, n.d.). In the construction, we notice that the distance from a vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.
The circumcenter is the point from which the triangle’s vertices are always equidistant. To construct a circumcenter of a triangle, construct a circle such that all three of the triangle’s vertices lie on the circle, and that circle’s center is the circumcenter of the triangle. Notice that the circumcenter is not always located inside the triangle (Dunbar, n.d.). Specifically, we see that the circumcenter is located outside of obtuse triangles.
The orthocenter is the point of intersection of the altitudes of a triangle. Recall that an altitude is the line extending from a vertex that is perpendicular to the opposite side. Like the circumcenter, notice that the orthocenter is located outside obtuse triangles (Dunbar, n.d.).
The incenter is the center of a circle inscribed within a triangle and is constructed by finding the intersection point of the angle bisectors in a triangle (Dunbar, n.d.).
Now that we have this understanding of how to locate the various centers of a given triangle, we must understand the concept of length and distance in the context of two points on a plane (or locations on a map). To solve the given problem of finding the most economical stadium location, it is intuitive to reason that the shorter the road, the cheaper the cost to build it. Since it would be impossible to evaluate every possible pathway from one point to another, we need only look at the most direct. In this way, two primary routes will need to be examined from each city to each center: a straight shot (paved with all new roads) or an indirect route that utilizes existing roads to minimize the distance of newly paved roads. This is necessary because the cost of paving new roads ($125,000/mi.) is significantly more expensive than repaving existing roads ($50,000/mi.).
In the following construction, one can investigate to discover that the shortest possible distance between points A and B is a straight line (colored blue), which represents a new road paved between the two locations. However, in order to maximize use of an existing road (line BC), we need to know the shortest distance from point A to line BC. In the construction, the pink line connecting A to line BC is perpendicular to BC. Comparing this line with the orange line, one discovers that the way to minimize the distance of newly paved road is to follow the perpendicular (pink) path from A to line BC.
With this background knowledge of triangle centers and distances between points, continue to the next page to apply these concepts to find the most economical stadium location.