Title
The Third Side
Problem Statement
The lengths of two sides
of a triangle are 5 and 11. If the length of the third side is also a whole
number, what are the smallest and largest possible perimeters that the
triangle can have?
Problem setup
If the lengths of 2 sides of a
triangle are 5 and 11, what is the smallest and
the largest possible perimeters that the triangle can have, if the length
of the third side is also a whole number?
Plans to
Solve/Investigate the Problem
I will use Geometer’s Sketchpad
and my prior knowledge of the Triangle Inequality Theory to construct 2
circles with the radii of the 2 circles 11 units apart and the radius of
the 1^{st} circle will be 5 units in length. I will label the line segment formed
by joining the radii as line segment AB. I will label the radius of the 1^{st}
circle as line segment AC. I
will formulate a 3 side by creating a line segment between points B and C
and label it line segment BC. Then
I will animate point C to see how the length of the line segment changes as
the point travels around the circle.
Investigation/Exploration
of the Problem
Using Geometer’s Sketchpad
I constructed 2 circles. The radii
of the 2 circles were situated 11 units apart. The radius of the 1^{st} circle
was 5 units in length. I labeled
the line segment formed by joining the radii as line segment AB. I labeled the radius of the 1^{st}
circle as line segment AC. I formulated
a 3rd side by creating a line segment between points B and C and labeled it
line segment BC. Then I animated
point C to see how the length of the line segment changed as the point
traveled around the circle. I
determined, if you are using whole numbers, that the length of line segment
will be greater than 6 units but less than 16 units. The smallest perimeter will thus be 23
units and the greatest perimeter will be 31 units.
Extensions of the Problem
What if the lengths of 2 sides
of a triangle are 6 and 112, what is the smallest and the largest possible
perimeters that the triangle can have, if the length of the third side is
also a whole number?
Author & Contact
Insert name and contact information.
lparsons@rockdale.k12.ga.us
