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Let's take a look at our triangle:
Now, using the law of sines, we'll get that:
[tex]\begin{gathered} \frac{b}{\sin B}=\frac{c}{\sin C} \\ \text{and} \\ \frac{b}{\sin B}=\frac{a}{\sin A} \end{gathered}[/tex]We know that the sum of the three interior angles of a trianlge is 180°. Therefore, we can conclude that:
[tex]\angle B=50[/tex]Now. solving for a and c,
[tex]\begin{gathered} \frac{b}{\sin B}=\frac{c}{\sin C}\rightarrow c=\frac{b\sin C}{\sin B}\Rightarrow c=7.36 \\ \text{and} \\ \frac{b}{\sin B}=\frac{a}{\sin A}\rightarrow a=\frac{b\sin A}{\sin B}\Rightarrow a=2.68 \end{gathered}[/tex]This way, we know the lenght of the three sides of the triangle:
[tex]\begin{gathered} a=2.68 \\ b=6 \\ c=7.36 \end{gathered}[/tex]Now, we can calculate the area of such triangle using Heron's formula:
[tex]A=\sqrt[]{s(s-a)(s-b)(s-c)}[/tex]Where:
[tex]s=\frac{a+b+c}{2}[/tex]Using this, and the sides we've just calculated, we'll have that:
[tex]s=\frac{2.68+6+7.36}{2}\Rightarrow s=8.02[/tex]Thereby,
[tex]\begin{gathered} A=\sqrt[]{8.02(8.02-2.68)(8.02-6)(8.02-7.36)} \\ \Rightarrow A=7.56 \end{gathered}[/tex]We can conclude that the area of the triangle is 7.56 square feet
