InterMath
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, then each team has to run the same distance to get to the flag.
Where should the flag be positioned for the game to be fair? Describe how you found
the position.
Problem setup
In order for the position of the flag to be fair, we need to find the point which is an equal
distance from each vertex of the triangle.
Plans to Solve/Investigate the Problem
I will use the Geometer’s Sketchpad to construct a scalene triangle which is a triangle
with no equal sides. Then I will find the circumcenter which is the point of intersection of
the perpendicular bisectors of the sides of a given triangle; the center of the circle
circumscribed about a given triangle. The circumcenter will be the point which is an
equal distance from all vertices of the triangle. I will then create line segments from the
circumcenter to each of my vertices and measure the length to make sure they are all
the same. The circumcenter will be the place where the flag should be placed in the
field
Investigation/Exploration of the Problem
In order to discuss how I went about solving this problem I will do it in a step by step
process:
1. First, I will construct a scalene triangle within a circle. I will do this by using GSP.
I will use the compass tool to draw a circle. Then I will put 3 points on the circle
and connect those points using line segments. I will then measure each of my
line segments to be sure that none of them have the same measurement. To
have a scalene triangle, I must be sure none of my sides are equal.
Team A
Team B
Team C
m Team BTeam A = 5.00 cm
m Team CTeam B = 4.40 cm
m Team ATeam C = 3.45 cm
2. Next, I will find the midpoint of the line each line segment. I will then create
perpendicular lines through each midpoint on the lines. This will serve as the
perpendicular bisectors of each side of my triangle.
Team A
Team B
Team C
3. Next, I will plot a point where each of the perpendicular bisectors intersect each
other. This will serve as the circumcenter of the triangle, which is the center of
the circumscribed circle around the triangle (definition from intermath website).
Note: I labeled the circumcenter C.
Team A
Team B
C
Team C
4. Since we know that the circumcenter (point C) is the center of the circle, we can
create line segments from the center point to each vertex of the triangle. Since each
vertex touches the outside edge of the circle, we can draw a segment from the center
point to each vertex of the circle. These segments will serve as radii of the circle and
therefore should all be equal.
Team A to fl ag = 2.56 cm
Team B to fl ag = 2.56 cm
Team A
Team B
Team C to fl ag = 2.56 cm
C
Team C
5. The red line represents the distance from team A to the flag. The blue line from
team C to the flag and the green line represent the distance from team B to the
flag. Each distance measures 2.56cm and therefore is all equal and all serve as
radii of the circle and therefore are congruent. Therefore, the flag should be
placed at the circumcenter of the circle (point C) so that the distance to the flag is
the same for each team.
Extensions of the Problem
The extension, I will explore comes from the intermath website. The question is: How
would the position of the flag change if the field has n-sides and the teams have to start
somewhere on the perimeter of the field?
First, I will begin by exploring this concept by constructing a hexagon shaped field in
GSP, however any n-sided polygon could work. I will use the same principle as I did in
my initial problem except this time the circle will be inscribed within the polygon. For
this to work, I must have a regular polygon where all sides are equal and congruent as
well as the angles. The polygon must be regular because if it’s not then the polyon will
not touch all sides of the circle. To find the center of the inscribed circle (the incenter), I
must find the point of intersection of the angle bisectors of the interior angles of the
polygon. The definition of inscribed circle states that the inscribed circle is tangent to
the sides of the polygon and therefore the distance from the incenter to the sides are
radii of the same circle and therefore are congruent. Therefore, the flag should be
placed at the incenter of the circle.
A
Team A F
m Team AFLAG = 2 cm
Team F
Team B
m Team BFLAG = 2 cm
m Team CFLAG = 2 cm
B
FLAG
E
m Team DFLAG = 2 cm
m Team EFLAG = 2 cm
Team E
C
Team C
Team D
m Team FFLAG = 2 cm
D
As you can see, the distance from each team to the flag is the same. Therefore, the
flag should be placed at the incenter. The line segments that represent each team
serve as radii of the circle and therefore are all congruent. So in conclusion, to solve for
any polygon, simply inscribe a regular polygon within a circle and find the perpendicular
bisectors and incenter of the circle.
Author & Contact
Carla McNeely, Middle Grades Education student, concentrating in English/Language
Arts and Math. I am currently a junior at Georgia College and State University.
carlalynnmc@yahoo.com
Link(s) to resources, references, lesson plans, and/or other materials
http://intermath.coe.uga.edu/dictnary