Question

Um triângulo regular e um hexágono regular possuem áreas iguais. Se o lado do hexágono tem medidas iguais a 4 m, então, o perímetro do triângulo será, em m, de: A) 4 B) 12 C) 4√6 D) 12√6

Answer 1

$\large\boxed{\begin{array}{l}\underline{\sf \acute Area\,do\,hex\acute agono}\\\rm A_{hex}=\dfrac{3\ell^2\sqrt{3}}{2}\\\underline{\sf\acute Area\,do\,tri\hat angulo\,regular}\\\rm A_{\triangle}=\dfrac{l^2\sqrt{3}}{4}\\\rm no\,hex\acute agono\,\ell=4\,m\\\rm logo\\\rm A_{hex}=\dfrac{3\cdot4^2\sqrt{3}}{2}=\dfrac{3\cdot16\sqrt{3}}{2}=24\sqrt{3}\\\rm para\,o\,tri\hat angulo\,teremos\\\rm \dfrac{l^2\diagup\!\!\!\!\!\sqrt{3}}{4}=24\diagup\!\!\!\!\!\sqrt{3}\\\\\rm l^2=24\cdot4\\\rm l^2=96\end{array}}$

$\large\boxed{\begin{array}{l}\rm l=\sqrt{96}\\\rm l=\sqrt{16\cdot6}\\\rm l=4\sqrt{6}\,m\\\rm representando\,por\,2p\\\rm o\,per\acute imetro\,teremos\\\rm 2p=3\cdot l\\\rm 2p=3\cdot4\sqrt{6}\\\rm 2p=12\sqrt{6}\,m\\\huge\boxed{\boxed{\boxed{\boxed{\rm\green{\dagger}\red{\maltese}~\blue{alternativa~D}}}}}\end{array}}$