Geometry

The area of trapezoids ABEC, ABFD, and ABFC are 133, 140, and 161 square feet, respectively.

Quadrilateral ABED is a rectangle.

The length of segment DE is 8 feet.

What is the length of segment CF?

\(\text{Let $A_1= ABEC = 133$} \\ \text{Let $A_2= ABFD = 140$} \\ \text{Let $A = ABFC = 161$} \\ \text{Let $A_{\text{rectangle}}=A_\square = 8 AD $} \\ \text{Let $x = CF$} \)

\(\begin{array}{|rcll|} \hline A_1 + A_2 &=& A + A_\square \\ 133+140 &=& 161 + A_\square \\ A_\square &=& 133+140- 161 \\ \mathbf{A_\square} &=& \mathbf{112} \quad | \quad A_\square = 8 AD \\ 8 AD &=& 112 \\ AD &=& \dfrac{112}{8} \\ \mathbf{AD} &=& \mathbf{14} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline A &=& \dfrac{(8+x)}{2}*AD \\ 161 &=& \dfrac{(8+x)}{2}*14 \\ 161 &=& (8+x)*7 \\ 23 &=& 8+x \\ x &=& 23-8 \\ \mathbf{x} &=& \mathbf{15} \\ \hline \end{array}\)

The length of segment CF is 15