C is the midpoint of Segment AB, D is the midpoint of Segment AC, E is the midpoint of Segment AD, F is the midpoint of Segment ED, G is the midpoint of Segment EF, and H is the midpoint of Segment DB. If DC= 16, what is GH


To solve this problem, we need to analyze the segments and midpoints given in the problem. Let's define the length of each segment and find the relationship between them step by step.

### Step 1: Assign Lengths to Segments
1. Let \( AB = x \).
2. Since \( C \) is the midpoint of \( AB \), \( AC = \frac{x}{2} \) and \( CB = \frac{x}{2} \).
3. Since \( D \) is the midpoint of \( AC \), \( AD = \frac{AC}{2} = \frac{x}{4} \) and \( DC = \frac{AC}{2} = \frac{x}{4} \).
4. Since \( E \) is the midpoint of \( AD \), \( AE = \frac{AD}{2} = \frac{x}{8} \) and \( ED = \frac{AD}{2} = \frac{x}{8} \).
5. Since \( F \) is the midpoint of \( ED \), \( EF = \frac{ED}{2} = \frac{x}{16} \) and \( FD = \frac{ED}{2} = \frac{x}{16} \).
6. Since \( G \) is the midpoint of \( EF \), \( EG = \frac{EF}{2} = \frac{x}{32} \) and \( GF = \frac{EF}{2} = \frac{x}{32} \).
7. Since \( H \) is the midpoint of \( DB \), \( DH = \frac{DB}{2} \).

### Step 2: Given Information
We know that \( DC = 16 \).

Since \( DC = \frac{x}{4} \), we have:
\[
\frac{x}{4} = 16
\]
\[
x = 64
\]

### Step 3: Find \( GH \)
We know that:
\[
EF = \frac{x}{16} = \frac{64}{16} = 4
\]
\( G \) is the midpoint of \( EF \), so:
\[
EG = GF = \frac{EF}{2} = \frac{4}{2} = 2
\]

\( H \) is the midpoint of \( DB \). To find \( DB \), consider:
\[
DB = DB = AB - AD - DB = 64 - 16 = 48
\]
Thus, \( H \) is the midpoint of \( DB \), so:
\[
DH = HB = \frac{48}{2} = 24
\]

Now, we find \( GH \), the difference between \( G \) and \( H \):
Since \( GH = EG - DH \),
\[
GH = 2 - 24 = 22
\]

Therefore, \( GH \) is \( 2 \).

C is the midpoint of Segment AB, D is the midpoint of Segment AC, E is the midpoint of Segment AD, F is the midpoint of Segment ED, G is the midpoint of Segment EF, and H is the midpoint of Segment DB. If DC = 16, what is GH?

Explore the solution to finding the segment GH, given midpoints on a line segment and a known length DC = 16. This problem involves key geometric concepts such as midpoints and line segments, suitable for students in Grades 8-10.

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