Let's go through the problem step-by-step, filling in the missing values and completing the statements.
1. [tex]\( A, B, C, \)[/tex] and [tex]\( D \)[/tex] are consecutive points on a line such that [tex]\( AB = BC = CD = 6 \)[/tex] cm.
- Given [tex]\( AB = 6 \)[/tex] cm.
2. Find the midpoints [tex]\( M \)[/tex] and [tex]\( N \)[/tex] of segments [tex]\( \overline{AB} \)[/tex] and [tex]\( \overline{CD} \)[/tex], respectively.
[tex]\[
\begin{tabular}{|l|l|}
\hline
Statement & Reason \\
\hline
$AM = BM =\frac{ AB }{2}=\frac{6}{2} = 3$ cm & Definition of midpoint \\
\hline
$CN =\frac{ CD }{2}= \frac{6}{2} = 3 \text { cm }$ & Definition of midpoint \\
\hline
$MN = MB + BC + CN$ & Total distance includes midpoint to endpoint of AB, distance from B to C, and midpoint from endpoint of CD \\
\hline
$MN = 3 \text{ cm } + 6 \text{ cm } + 3 \text{ cm } = 12$ cm & Algebra \\
\hline
\end{tabular}
\][/tex]
To summarize:
- The distance [tex]\( AM = BM = \frac{AB}{2} = \frac{6}{2} = 3 \)[/tex] cm.
- Similarly, the distance [tex]\( CN = \frac{CD}{2} = \frac{6}{2} = 3 \)[/tex] cm.
- The distance [tex]\( MN = MB + BC + CN = 3 \text{ cm } + 6 \text{ cm } + 3 \text{ cm } = 12 \)[/tex] cm.
Thus, the distance between the midpoints [tex]\( M \)[/tex] and [tex]\( N \)[/tex] is [tex]\( 12 \)[/tex] cm.
