Chapter 12.4B, Problem 21A

a. In the figure, $ABCD$ is a trapezoid, $M$ is the midpoint of $\overline{AB}$. Through $M$, a line parallel to the bases has been drawn, intersecting $\overline{CD}$ at $N$. (i) Explain why $N$ must be the midpoint of $\overline{CD}$, and (ii) express $MN$ in terms of $a$ and $b$, the lengths of the parallel sides of the trapezoid.

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b. Denote $MN$ by $c$. Use your answer in part (a) to show that $b,c,a$ form an arithmetic sequence.

c. In the trapezoid $ABCD$, the lengths of the bases are $a$ and $b$ as shown. Side $\overline{AB}$ has been divided into $9$ congruent segments. Through the endpoints of the segments, lines parallel to $\overline{AD}$ have been drawn. In this way, $8$ new segments connecting the sides $\overline{AB}$ and $\overline{CD}$ have been created. Show that the sequence of $10$ terms, starting with $b$, proceeding with lengths of the parallel segments, and ending with $a$, is an arithmetic sequence and find the sum of the sequence in terms of $a$ and $b$.