Question

Points A, B, E and F are midpoints of $\overline{XC},\overline{XD},\overline{YC}$and $\overline{YD}$. Complete.

If $\text{AB}=\text{k}$, then $CD=?$ and $\text{EF}=?$ .

Short answer

$\text{CD}=\mathbf{2}\text{k}$ and $\text{EF}=\text{k}$ .

Step-by-step solution

Step 1. Consider the diagram.

Here, A, B, E and F are the midpoints of sides $\overline{XC},\overline{XD},\overline{YC}$and $\overline{YD}$ respectively.

and $AB=k$

Step 2. Show the calculation.

Consider the triangle $△XCD$.

Here, points A and B are the midpoints 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 midpoints 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 $\overline{CD}$.

$\begin{array}{c}EF=\frac{1}{2}CD\\ =\frac{1}{2}\times 2k\\ =k\end{array}$

Step 3. State the conclusion.

Therefore, $\overline{CD}=2k$ and $\overline{EF}=k$.