$\overline{\mathrm{MN}}$is the median of trapezoid $\mathbf{ZOID}$.

If$\mathbf{ZO}\mathbf{=}\mathbf{8}$ , then$\mathrm{TN}=\underset{\_}{?}$ .

The given diagram is:

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.

It is being given that$\overline{MN}$ is the median of trapezoid zoid.

From the given diagram it can be noticed the sidesZO and DL are parallel sides, therefore the bases of the trapezoid ZOID are ZOand DL.

Therefore, by using the theorem 5-19 it can be noticed that the median is parallel to the basesandZO and DLhas a length equal to the average of the base lengthsZO and DL.

Therefore,$\overline{MN}\parallel \overline{ZO}$.

Therefore, it can be said that $\overline{TN}\parallel \overline{ZO}$.

As, is the median of trapezoid ZOID, therefore it can be said that is the median of the triangle .

The theorem 5-11 states that the segments that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.

Therefore, as is the median of the triangle , therefore by using the theorem 5-11 it can be said that$\overline{TN}\parallel \overline{ZO}$and$\overline{TN}=\frac{1}{2}\left(\overline{ZO}\right)$.

Therefore, it can be obtained that:

$\begin{array}{c}\overline{TN}=\frac{1}{2}\left(\overline{ZO}\right)\\ =\frac{1}{2}\left(8\right)\\ =4\end{array}$

Therefore the length ofTN is 4.

If $ZO=8$, then$TN=4$ .