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# Geometry

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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?

https://latex.artofproblemsolving.com/0/f/f/0ff801c98054c45ce4a7974db31d7c1c0fe19fd8.png

Dec 18, 2019

#1
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The area of trapezoids ABEC, ABFD, and ABFC are 133, 140, and 161 square feet, respectively.

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

Dec 18, 2019