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Title
Capture the Flag

Problem Statement
Three teams: A, B, and C, each start from a vertex of a scalene triangular field. Their goal is to be the first team to grab the flag that is located inside the triangular field. If the game is fair, each team has to run the same distance to get the flag.

Where should the flag be positioned to be fair?

Describe how you found the position.

Problem setup
To make the game fair, each team must travel an equal distance to a flag somewhere inside the field. Each team will start from a vertex of the scalene triangular field and none of the sides of the field are congruent. This investigation is similar to others because it causes you to discover the properties of scalene triangles through the investigation.

Plans to Solve/Investigate the Problem
To solve this problem I plan to try to visualize different scalene triangles and think about how to make each vertex an equal distance from some point inside the triangle. I plan to use Geometer’s Sketch pad to construct a scalene triangle whose vertices lie on a circle. For this approach to work it is necessary for the center of the circle to be inside of the triangle, and make sure that none of the sides are congruent.

Investigation/Exploration of the Problem
If you want to make three points an equal distance from another in the interior of the triangle to make the three teams an equal distance from the flag, a circle would be a helpful tool to use. Start by constructing the circle. The size of the circle should not be a central focus in this investigation, only that the vertices must be an equal distance from the center so that they lie on the circle. These points must be constructed so that the vertices are not an equal distance from each other. The center of the circle lies in the interior of the triangle formed by the segments connecting the vertices. To efficiently produce the desired results, construct segments connecting the vertices and calculate the distance between them. Once the segments have been constructed and measured you will be able to drag the points so that they are not an equal distance from each other and that the center of the circle, or the flag, is in the interior of the triangle. Using the circle approach ensures that the starting points for each team will be an equal distance from the flag. The two conditions that must be met by dragging the points are being sure that the starting positions are not an equal distance from each other and that the flag is positioned inside the playing field.

Extensions of the Problem
How would the position of the flag change if the field had n-sides and the teams had to start somewhere on the perimeter of the field?

Using the circle to solve the investigation, will any scalene triangle work?

1. One triangle that would not work is one that has its longest side equal to the diameter of the circle. This triangle would put the flag on the perimeter of the field and not in the interior.

Author & Contact
Nathan Little
nlittle@aug.edu

Link(s) to resources, references, lesson plans, and/or other materials
My constructions
Intermath

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