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From the diagram provided in the question, we have the following triangle:
We have the following angle and side measures:
[tex]\begin{gathered} z=80\degree \\ y=40\degree \\ G=12\text{ feet } \\ x=180-80-40=60\degree\text{ (Sum of angles in a triangle)} \end{gathered}[/tex]Recall the Sine Rule:
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]Comparing to our triangle, we have the sine rule applied as:
[tex]\frac{L}{\sin x}=\frac{G}{\sin z}=\frac{S}{\sin y}[/tex]Length of Longer Wire (L):
[tex]\frac{L}{\sin x}=\frac{G}{\sin z}[/tex]Substituting the given values, we have:
[tex]\begin{gathered} \frac{L}{\sin60}=\frac{12}{\sin 80} \\ L=\frac{12\sin 60}{\sin 80} \\ L=10.55\text{ feet} \end{gathered}[/tex]The length of the longer wire is approximately 10.6 feet.
Length of Shorter Wire (S):
[tex]\frac{G}{\sin z}=\frac{S}{\sin y}[/tex]Substituting known values, we have:
[tex]\begin{gathered} \frac{12}{\sin80}=\frac{S}{\sin 40} \\ S=\frac{12\sin 40}{\sin 80} \\ S=7.83\text{ feet} \end{gathered}[/tex]The length of the shorter wire is approximately 7.8 feet.
