say "a" is the length of the first piece, we'll first off, change the mixed fraction to "improper" and add the pieces, keeping in mind that, all 3 pieces add up to 10 feet.
[tex]\bf \stackrel{mixed}{4\frac{1}{2}}\implies \cfrac{4\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{9}{2}}\\\\
-------------------------------[/tex]
[tex]\bf \stackrel{\textit{first piece}}{a}+\stackrel{\textit{second piece}}{\cfrac{a}{2}}+\left(\stackrel{\textit{third piece}}{\cfrac{a}{2}~+~\cfrac{9}{2}}\right)=10\impliedby
\begin{array}{llll}
\textit{multiplying both sides by }\\
\textit{the LCD of 2}
\end{array}
\\\\\\
2\left( a+\cfrac{a}{2}+\cfrac{a}{2}+\cfrac{9}{2} \right)=2(10)\implies 2a+a+a+9=20
\\\\\\
4a=11\implies a=\cfrac{11}{4}\implies a=2\frac{3}{4}[/tex]
