<p>Let <img alt="${\triangle ABC}$" height="18" src="https://latex.artofproblemsolving.com/b/1/7/b17d611df8ee5251331fad2e6b688b8382a9d5ba.png" width="61" /> be a right triangle with <img alt="$\angle A = 90^\circ$" height="15" src="https://latex.artofproblemsolving.com/a/1/4/a147cb83e3b9fdf390e7d2cf520aa0ba7296fb2c.png" width="78" /> and <img alt="$BC = 38.$" height="14" src="https://latex.artofproblemsolving.com/1/8/6/186701f723178c658335843353a865df81499548.png" width="78" /> There exist points <img alt="$K$" height="12" src="https://latex.artofproblemsolving.com/d/f/b/dfb064112b6c94470339f6571f69d07afc1c024c.png" width="16" /> and <img alt="$L$" height="12" src="https://latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png" width="12" /> inside the triangle such<img alt="\[AK = AL = BK = CL = KL = 14.\]" height="13" src="https://latex.artofproblemsolving.com/c/b/3/cb31cd53c432faa251c220830b62d5a30ee8f6df.png" width="285" />The area of the quadrilateral <img alt="$BKLC$" height="14" src="https://latex.artofproblemsolving.com/1/8/7/187525daff20081bd8b5da8081edae998ee7a625.png" width="60" /> can be expressed as <img alt="$n\sqrt3$" height="21" src="https://latex.artofproblemsolving.com/7/8/a/78aa363d86b80f7e89f310a95cd99020a4461a57.png" width="37" /> for some positive integer <img alt="$n.$" height="8" src="https://latex.artofproblemsolving.com/b/d/9/bd923e41635af40a1abe58f230732f2280b0723a.png" width="14" /> Find <img alt="$n.$" height="8" src="https://latex.artofproblemsolving.com/b/d/9/bd923e41635af40a1abe58f230732f2280b0723a.png" width="14" /></p>

Question No: 333

Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such $AK = AL = BK = CL = KL = 14.$ The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

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Tags: Algebra Geometry

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