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. Show the work you you used to prove if Lena can or cannot use these pieces of fencing to build her triangular garden.
(my answer was yes)
Be sure to show all the steps in your work.

Answer 1

Use the triangle inequality: three lengths a, b, c can form a (nondegenerate) triangle exactly when each side is less than the sum of the other two:
a < b + c, b < a + c, and c < a + b.

Here a = 8, b = 6, c = 4.

Check each inequality:
1. 8 < 6 + 4 → 8 < 10 (true)
2. 6 < 8 + 4 → 6 < 12 (true)
3. 4 < 8 + 6 → 4 < 14 (true)

All three inequalities hold strictly, so Lena can form a triangle (not a straight line).

As a quick extra check, compute the area with Heron’s formula:
s = (8 + 6 + 4)/2 = 9
Area = sqrt[s(s−8)(s−6)(s−4)] = sqrt[9·1·3·5] = sqrt(135) = 3√15 ≈ 11.62 > 0,
confirming a nondegenerate triangle.

Classification: 8^2 = 64 and 6^2 + 4^2 = 36 + 16 = 52, so 64 > 52; the triangle is obtuse (the angle opposite the 8-ft side is obtuse).

Conclusion: Yes — Lena can build a triangle with sides 8 ft, 6 ft, and 4 ft.