Chapter 5.5, Problem 18WE

To determine

To find: The value of CF .

Answer to Problem 18WE

The value of CF is 30.

Explanation of Solution

Given:

The triangle is shown below:

  

  $\begin{array}{l}TA=AB=BC\\ TD=DE=EF\end{array}$

  $\begin{array}{l}AD=x+y\\ BE=20\\ CF=4x-y\end{array}$

Calculation:

Consider a triangle TBE ,

AD is the midsegment of the triangle TBE .

According to the midsegment theorem, the length of midsegment is half of base of triangle.

Hence,

  $\begin{array}{c}AD=\frac{BE}{2}\\ x+y=\frac{20}{2}\\ x+y=10\\ y=10-x\mathrm{......}\left(1\right)\end{array}$

Consider a trapezoid CFDA ,

  $\begin{array}{l}BC=AB\\ \text{And},\\ EF=DE\end{array}$

Thus, BF is the midsegment of the trapezoid CFDA .

According to the midsegment theorem, the length of midsegment is half of sum of parallel sides of trapezoid.

Hence,

  $\begin{array}{c}BE=\frac{AD+CF}{2}\\ AD+CF=2BE\\ x+y+4x-y=2\left(20\right)\\ 5x=40\\ x=8\end{array}$

Substitute 8 for x in equation (1) as follows:

  $\begin{array}{c}y=10-8\\ =2\end{array}$

Side CF is calculated as follows:

  $\begin{array}{c}CF=4x-y\\ =4\left(8\right)-2\\ =30\end{array}$

Thus, the value of CF is 30.