Intermath | Workshop Support

 Write-up

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 1st 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 1st 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 1st 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 1st 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