Chapter 5.5, Problem 17WE

To determine

To find: The values of x and y .

Answer to Problem 17WE

The values of x and y are 9 and 15 respectively.

Explanation of Solution

Given:

The triangle is shown below:

  

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

  $\begin{array}{l}AD=x+3\\ BE=x+y\\ CF=36\end{array}$

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,

  $\begin{array}{c}AD=\frac{BE}{2}\\ x+3=\frac{x+y}{2}\\ 2x+6=x+y\\ x=y-6\mathrm{......}\left(1\right)\end{array}$

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,

  $\begin{array}{c}BE=\frac{AD+CF}{2}\\ AD+CF=2BE\\ x+3+36=2\left(x+y\right)\\ x+39=2x+2y\\ x=39-2y\mathrm{......}\left(2\right)\end{array}$

Solve equations (1) and (2) as follows:

  $\begin{array}{l}y-6=39-2y\\ 3y=45\\ y=15\end{array}$

Substitute 15 for y in equation (1) as follows:

  $\begin{array}{c}x=15-6\\ =9\end{array}$

Thus, the values of x and y are 9 and 15 respectively.