Chapter 5.3, Problem 2WE

To determine

To find: the values of $CD$ and $EF$ .

Answer to Problem 2WE

  $CD=2k$

  $EF=k$

Explanation of Solution

Given:

The diagram:

Points A, B, E and F are the mid points of $\overline{XC}$ , $\overline{XD}$ , $\overline{YC}$ and $\overline{YD}$ .

  $AB=k$

Consider the figure shown below:

Consider the triangle $△XCD$ .

Here, points A and B are the mid points of $\overline{XC}$ and $\overline{XD}$ .

So, by theorem 5-11 part (2), the side $\overline{AB}$ is half as long as the third side $\overline{CD}$ .

  $AB=\frac{1}{2}CD$

So,

  $\begin{array}{c}CD=2AB\\ =2k\end{array}$

Consider the triangle $△YCD$ .

Here, points E and F are the mid points of $\overline{YC}$ and $\overline{YD}$ .

So, by theorem 5-11 part (2), the side $\overline{EF}$ is half as long as the third side $\overline{CD}$ .

  $\begin{array}{c}EF=\frac{1}{2}CD\\ =\frac{1}{2}\left(2k\right)\\ =k\end{array}$