Question

UNIFENAS 2019: Sendo dados os pontos A( -2; -3), B( 4; 1) e C(-4;3) de um triângulo, determine a área delimitada por estes pontos.​

Answer 1

✅ Após resolver os cálculos, concluímos que a área do referido triângulo é:

        $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf S_{\triangle} = 22\:u^{2}\:\:\:}}\end{gathered} }$

Sejam os pontos:

              $\Large\begin{cases} A(-2, -3)\\B(4, 1)\\C(-4, 3)\end{cases}$

Já que nos foi dado apenas os vértices do triângulo, podemos calcular sua área, obtendo a metade do módulo do determinante da matriz M se, e somente se a matriz "M" for:

      $\Large\displaystyle\begin{gathered} M = \begin{bmatrix} -2 & -3 & 1\\4 & 1 & 1\\-4 & 3 & 1\end{bmatrix}\end{gathered}$

Então, temos:

   $\Large\displaystyle\begin{gathered} S_{\triangle} = |\det M | / 2\end{gathered}$

            $\Large\displaystyle\begin{gathered} = \left|\begin{vmatrix} -2 & -3 & 1\\4 & 1 & 1\\-4 & 3 & 1\end{vmatrix}\right| / 2\end{gathered}$

             $\Large\displaystyle\begin{gathered} = \left|\begin{vmatrix} 1 & 1\\3 & 1\end{vmatrix}\cdot(-2) - \begin{vmatrix} 4 & 1\\-4 & 1\end{vmatrix}\cdot(-3) + \begin{vmatrix}4 & 1\\ -4 & 3\end{vmatrix}\cdot1\right| / 2\end{gathered}$

              $\large\displaystyle\begin{gathered} = \left|(1 - 3)\cdot (-2) - (4 + 4)\cdot(-3) + (12 + 4)\cdot1\right| / 2\end{gathered}$

              $\Large\displaystyle\begin{gathered} = |(-2)\cdot(-2) - 8\cdot(-3) + 16\cdot1| / 2\end{gathered}$

              $\Large\displaystyle\begin{gathered} = |4 + 24 + 16| / 2\end{gathered}$

              $\Large\displaystyle\begin{gathered} = |44| / 2\end{gathered}$

              $\Large\displaystyle\begin{gathered} = 44/2\end{gathered}$

              $\Large\displaystyle\begin{gathered} = 22\end{gathered}$

✅ Portanto, a área do triângulo é:

      $\Large\displaystyle\begin{gathered} S_{\triangle} = 22\:u^{2}\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$