<p>For positive real numbers <img alt="$s$" height="8" src="https://latex.artofproblemsolving.com/f/3/7/f37bba504894945c07a32f5496d74299a37aa51c.png" width="8" />, let <img alt="$\tau(s)$" height="18" src="https://latex.artofproblemsolving.com/7/9/6/796fd2efea8c821b0740238f661aba21bf3db31d.png" width="31" /> denote the set of all obtuse triangles that have area <img alt="$s$" height="8" src="https://latex.artofproblemsolving.com/f/3/7/f37bba504894945c07a32f5496d74299a37aa51c.png" width="8" /> and two sides with lengths <img alt="$4$" height="12" src="https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png" width="9" /> and <img alt="$10$" height="13" src="https://latex.artofproblemsolving.com/f/c/6/fc606f7f1e530731ab4f1cc364c01dc64a4455ee.png" width="17" />. The set of all <img alt="$s$" height="8" src="https://latex.artofproblemsolving.com/f/3/7/f37bba504894945c07a32f5496d74299a37aa51c.png" width="8" /> for which <img alt="$\tau(s)$" height="18" src="https://latex.artofproblemsolving.com/7/9/6/796fd2efea8c821b0740238f661aba21bf3db31d.png" width="31" /> is nonempty, but all triangles in <img alt="$\tau(s)$" height="18" src="https://latex.artofproblemsolving.com/7/9/6/796fd2efea8c821b0740238f661aba21bf3db31d.png" width="31" /> are congruent, is an interval <img alt="$[a,b)$" height="18" src="https://latex.artofproblemsolving.com/2/9/0/2905fb4cb148b1dc66f2cf1b1b0cee4353a594af.png" width="36" />. Find <img alt="$a^2+b^2$" height="16" src="https://latex.artofproblemsolving.com/c/f/6/cf669d58fbd0410ca860758e53606d06425a4d97.png" width="53" />.</p>

Question No: 361

For positive real numbers $s$ , let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$ .

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Tags: Algebra Geometry

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