Question

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? 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 whether Lena can construct a triangle using the pieces of fencing she has (8 feet, 6 feet, and 4 feet), we can apply the Triangle Inequality Theorem.

According to the Triangle Inequality 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 check this by comparing the three combinations of the sides.

Step 1: Check the Combinations

1. Check if \(8 + 6 > 4\): \[ 8 + 6 = 14 \quad \text{(which is greater than 4)} \] This condition is satisfied.

2. Check if \(8 + 4 > 6\): \[ 8 + 4 = 12 \quad \text{(which is greater than 6)} \] This condition is satisfied.

3. Check if \(6 + 4 > 8\): \[ 6 + 4 = 10 \quad \text{(which is greater than 8)} \] This condition is also satisfied.

Conclusion

Since all three conditions of the Triangle Inequality Theorem are satisfied:

1. \(8 + 6 > 4\)
2. \(8 + 4 > 6\)
3. \(6 + 4 > 8\)

Part 1 Answer

According to the Triangle Inequality Theorem, Lena will be able to build a triangle using the pieces of fencing she has.

Part 2 Answer

To summarize the proof:

1. Evaluate \(8 + 6 > 4\):

   • \(8 + 6 = 14 > 4\) ✔️

2. Evaluate \(8 + 4 > 6\):

   • \(8 + 4 = 12 > 6\) ✔️

3. Evaluate \(6 + 4 > 8\):

   • \(6 + 4 = 10 > 8\) ✔️

Since all conditions are satisfied, Lena can indeed create a triangular garden with the fencing pieces of 8 feet, 6 feet, and 4 feet.