Smallest Scalene Obtuse Triangle with Integer Sides and Area
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Theorem
The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$.
Proof
From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
- $A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Here we have:
\(\ds s\) | \(=\) | \(\ds \dfrac {4 + 13 + 15} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16\) |
Thus:
\(\ds A\) | \(=\) | \(\ds \sqrt {16 \paren {16 - 4} \paren {16 - 13} \paren {16 - 15} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {16 \times 12 \times 3 \times 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {2^4 \times 2^2 \times 3 \times 3 \times 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {2^6 \times 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 24\) |
This theorem requires a proof. In particular: It remains to be shown that it is the smallest. This can be done by exhaustion: the number of triples defining an obtuse scalene triangle are not that many. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$