The Degenerate Triangle Garden

Key idea. Check the triangle inequality before computing an area.

The two shorter sides sum to the longest:

55 + 62 = 117 .

A non-degenerate triangle needs the sum of any two sides to be strictly greater than the third. Here equality holds, so the three vertices are collinear. The third vertex lies on the 117-yard side, splitting it into pieces of 55 and 62 yards. The “triangle” is a straight segment and encloses no area.

Heron’s formula confirms this. The semi-perimeter is $s = \tfrac{1}{2}(55+62+117) = 117$ , so

\text{Area} = \sqrt{s(s-55)(s-62)(s-117)} = \sqrt{117\cdot 62\cdot 55\cdot 0} = 0 .

At $10 per square yard, the plot costs nothing.

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