InterMath | Workshop Support | Write Up Template
Title
Planting Trees
Problem Statement
A farmer wants to plant four trees, each one the same distance from any one of the other
three. Can the farmer do it? If so, how?
Problem setup
Using a triangular shape, is it possible to plant four trees and have each tree the exact same
distance apart any of the other trees? Explain and justify you answer.
Plans to Solve/Investigate the Problem
I predict that it is possible to plant four trees that are each an equal distance away from all of the
other trees. To solve this problem, I first plan to construct an equilateral triangle using GSP. I
will then find the angle bisector of each vertex on the triangle. I will then attempt to find a point
that will be equidistant from the remaining three points on the triangle.
Investigation/Exploration of the Problem
I first constructed an equilateral triangle. I created two points and connected those points with a
line segment. I then selected the line segment and rotated it around Point Tree 1 sixty degrees. I
rotated the segment because I know that the sum of all the angles of a triangle must total 180
degrees, and since the triangle is an equilateral, then all of the angles must be equal also. So, 180
degrees divided by three is 60 degrees per angle. I then connected the points Tree 2 and Tree 3
to form the base of the triangle. Once the triangle was constructed, I measured the distance of
each sides of the triangle. I found the distance of line segment (Tree 1)(Tree2), (Tree 2)(Tree 3),
and (Tree 1)(Tree 3) each to be 4.11 cm. From this information, one is able to prove that Tree 1,
Tree 2, and Tree 3 are all equal distances away from each other based on the properties of an
equilateral triangle.
Tree 1
Tree 3Tree 1 = 4.11 cm
Tree 1Tree 2 = 4.11 cm
Tree 3
Tree 2
Tree 2Tree 3 = 4.11 cm
Once I determined these three points, I then began my exploration to try to find a fourth point
that was an equal distance away from Tree 1, Tree 2, and Tree 3. I began by first exploring the
point in the center of the triangle. To create this point, I constructed an angle bisector though
angle (Tree1)(Tree2)(Tree3), angle (Tree2)(Tree3)(Tree1), and through angle
(Tree3)(Tree1)(Tree2). I then marked the intersection of the three angle bisectors, which I called
Tree 4 and measured the length of Tree 4 from the each of the other points already on the
triangle. Even though I found Tree 4 to be an equal distance from Tree 2, Tree 3, and Tree 1, the
distance that I measured was 2.37 cm. Because this distance does not measure to be 4.11cm,
Tree 4 is not at a distance equal from all three other points on the triangle.
Tree 1
Tree
E
4
Tree 3
Tree 2
Tree 3Tree 1 = 4.11 cm
Tree 1Tree 2 = 4.11 cm
Tree 2Tree 3 = 4.11 cm
Tree 3Tree 4 = 2.37 cm
Tree 2Tree 4 = 2.37 cm
Tree 1Tree 4 = 2.37 cm
Putting Tree 4 in the center of the triangle is my best guess of a possibility to make all points an
equal distance away from each other. As I have proven through measurement above, this is not
possible. Moreover, it is also evident that if Tree 4 were moved to any other point in the triangle
that this is a worse scenario because it will obviously be closer to one of the first three Trees than
another tree. Through this evidence and my construction of the triangle, I conclude that it is not
possible for a farmer to plant four trees, which are all an equal distance away from each other.
Extensions of the Problem
There were no extensions to the problem that I could find.
Author & Contact
Jennifer Woods
Jenn4507@yahoo.com