Lena is building a triangular garden. She has three pieces of fencing: one that is 8 feet long, one that is 6 feet long, and one that is 4 feet long. Part 1: Use the Triangle Inequality Theorem to check if Lena can create a triangle with these three pieces of fencing. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Will the lengths of fencing that Lena has work to create a triangle? (3 points) According to the Triangle Inequality Theorem, will Lena be able to build a trianlge using the pieces of fencing she has? _ yes or no

Question

Question 2 Part 2: Show the work you you used to prove if Lena can or cannot use these pieces of fencing to build her triangular garden. Be sure to show all the steps in your work. (3 points)

Answer 1

To determine if Lena can create a triangle with the three pieces of fencing (8 feet, 6 feet, and 4 feet), we will apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We will analyze each combination of the sides:

1. First combination:

   • 8 feet + 6 feet > 4 feet
   • 14 > 4 (True)

2. Second combination:

   • 8 feet + 4 feet > 6 feet
   • 12 > 6 (True)

3. Third combination:

   • 6 feet + 4 feet > 8 feet
   • 10 > 8 (True)

Since all three inequalities are true, Lena can indeed create a triangle with the pieces of fencing she has.

Part 1 Answer:

Yes, Lena will be able to build a triangle using the pieces of fencing she has.

Part 2 Work:

1. Evaluate the first inequality:

   • 8 + 6 > 4
   • 14 > 4 (True)

2. Evaluate the second inequality:

   • 8 + 4 > 6
   • 12 > 6 (True)

3. Evaluate the third inequality:

   • 6 + 4 > 8
   • 10 > 8 (True)

Since all three conditions of the Triangle Inequality Theorem are satisfied, it proves that Lena can build a triangle with the fencing she has.