✅ Tendo terminado os cálculos, concluímos que a área do referido triângulo é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf S_{\triangle} = 8\:m^{2}\:\:\:}}\end{gathered}$}[/tex]
Analisando a figura, concluímos que os vértices do respectivo triângulo são:
[tex]\Large\begin{cases} A (1, 3)\\B (3, 5)\\C(7, 1)\end{cases}[/tex]
Como temos apenas os vértices do triângulo então podemos aplicar conceitos de geometria analítica para calcular sua área. Para isso, basta calcular a metade do módulo do determinante da matriz "M".
Se a matriz "M" é:
[tex]\Large\displaystyle\text{$\begin{gathered}M = \begin{bmatrix} 1 & 3 & 1\\3 & 5 & 1\\7 & 1 & 1\end{bmatrix}\end{gathered}$}[/tex]
Então, temos:
[tex]\Large\displaystyle\text{$\begin{gathered} S_{\triangle} = \frac{|\det M|}{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered}= \begin{vmatrix}1 & 3 & 1\\3 & 5 & 1\\7 & 1 & 1 \end{vmatrix} / 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \left|\begin{vmatrix}5 & 1\\1 & 1 \end{vmatrix}\cdot 1 - \begin{vmatrix}3 & 1\\7 & 1 \end{vmatrix}\cdot3 + \begin{vmatrix} 3 & 5\\7 & 1\end{vmatrix}\cdot1\right|/2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \left\|(5 - 1)\cdot1 - (3 - 7)\cdot3 + (3 - 35)\cdot1\right\| / 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \left\|4\cdot1 - (-4)\cdot3 + (-32)\cdot1\right\| / 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = |4 + 12 - 32| /2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = |-16| / 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 16/2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 8\end{gathered}$}[/tex]
✅ Portanto, a área do triângulo é:
[tex]\Large\displaystyle\text{$\begin{gathered} S_{\triangle} = 8\:m^{2}\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
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[tex]\Large\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered}$}[/tex]
