Question

M and N are the midpoints of the legs of trapezoid EFGH. If bases$\overline{\mathrm{EF}}$and$\overline{\mathrm{HG}}$have lengths$\mathbf{2}\mathbf{r}\mathbf{+}\mathbf{s}$and $\mathbf{4}\mathbf{r}\mathbf{-}\mathbf{3}\mathbf{s}$, express the length of$\overline{\mathrm{MN}}$in terms of r and s .

Short answer

The length of$\overline{MN}$ in terms of r and s is $3r-s$.

Step-by-step solution

- Draw the possible diagram of the given problem.

The possible diagram of the given problem is:

It is being given that M and N are the midpoints of the legs of trapezoid EFGH.

It is also being given that $\overline{EF}=2r+s$and $\overline{HG}=4r-3s$.

- Write the theorem 5-19.

The theorem 5-19 states that the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths.

- Find the length ofMN¯  in terms of r and s.

Therefore, by using the theorem 5-19 it can be obtained that:

$\begin{array}{c}\overline{MN}=\frac{1}{2}(\overline{EF}+\overline{HG})\\ =\frac{1}{2}(2r+s+4r-3s)\\ =\frac{1}{2}(6r-2s)\\ =\frac{2}{2}(3r-s)\\ =3r-s\end{array}$

Therefore, the length of$\overline{MN}$in terms of r and s is $3r-s$.