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.

Question

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?
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)
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Answer 1

To determine whether Lena can create a triangle with her pieces of fencing using the Triangle Inequality Theorem, we can analyze the lengths of the sides (8 feet, 6 feet, and 4 feet) as follows:

Part 1:

According to the Triangle Inequality Theorem, we need to check the following three conditions:

1. The sum of the lengths of the first two sides must be greater than the length of the third side:

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

2. The sum of the lengths of the first and third sides must be greater than the length of the second side:

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

3. The sum of the lengths of the second and third sides must be greater than the length of the first side:

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

Conclusion for Part 1:

Since all three conditions are true, Lena can create a triangle using the pieces of fencing she has.

Part 2:

Steps for Proving:

1. Identify the lengths:

   • Side 1 = 8 feet
   • Side 2 = 6 feet
   • Side 3 = 4 feet

2. Apply the Triangle Inequality Theorem:

   • Check (1):
     \(8 + 6 > 4\)
     Calculation: \(14 > 4\) (True)

   • Check (2):
     \(8 + 4 > 6\)
     Calculation: \(12 > 6\) (True)

   • Check (3):
     \(6 + 4 > 8\)
     Calculation: \(10 > 8\) (True)

3. Final Result:
   Since all three inequalities hold true, Lena can build her triangular garden with the given pieces of fencing.

In summary, Lena can use the fencing to create a triangle as it satisfies the Triangle Inequality Theorem.