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

We are given a series of points, each defined as a midpoint, and our goal is to find the length of \( GH \), given that \( DC = 16 \).

Let’s analyze the situation step-by-step using a coordinate system for simplicity and clarity.

### Step 1: Assign coordinates
Let’s place point \( A \) at the origin: 
- \( A = (0, 0) \)
- Point \( B \) will then have coordinates \( B = (2x, 0) \) since \( C \) is the midpoint of \( AB \), and the distance between \( A \) and \( B \) is along the x-axis.

Thus, the coordinates of the midpoint \( C \) are:
\[
C = \left( \frac{0 + 2x}{2}, 0 \right) = (x, 0)
\]

### Step 2: Find other points
Next, we calculate the coordinates of the other midpoints based on the coordinate of their defining points:
- \( D \) is the midpoint of \( AC \), so its coordinates are:
\[
D = \left( \frac{0 + x}{2}, 0 \right) = \left( \frac{x}{2}, 0 \right)
\]
- \( E \) is the midpoint of \( AD \), so:
\[
E = \left( \frac{0 + \frac{x}{2}}{2}, 0 \right) = \left( \frac{x}{4}, 0 \right)
\]
- \( F \) is the midpoint of \( ED \), so:
\[
F = \left( \frac{\frac{x}{4} + \frac{x}{2}}{2}, 0 \right) = \left( \frac{3x}{8}, 0 \right)
\]
- \( G \) is the midpoint of \( EF \), so:
\[
G = \left( \frac{\frac{x}{4} + \frac{3x}{8}}{2}, 0 \right) = \left( \frac{5x}{16}, 0 \right)
\]
- \( H \) is the midpoint of \( DB \). We already know the coordinates of \( D = \left( \frac{x}{2}, 0 \right) \) and \( B = (2x, 0) \), so:
\[
H = \left( \frac{\frac{x}{2} + 2x}{2}, 0 \right) = \left( \frac{5x}{4}, 0 \right)
\]

### Step 3: Calculate the distance \( GH \)
Now we calculate the distance \( GH \) using the distance formula. Since all points lie on the x-axis, the distance between two points \( (x_1, 0) \) and \( (x_2, 0) \) is simply \( |x_2 - x_1| \).

The distance \( GH \) is:
\[
GH = \left| \frac{5x}{4} - \frac{5x}{16} \right| = \frac{5x}{4} - \frac{5x}{16} = \frac{20x - 5x}{16} = \frac{15x}{16}
\]

### Step 4: Use \( DC = 16 \) to find \( x \)
We are given that \( DC = 16 \). The distance between \( D = \left( \frac{x}{2}, 0 \right) \) and \( C = (x, 0) \) is:
\[
DC = |x - \frac{x}{2}| = \frac{x}{2}
\]
We know \( \frac{x}{2} = 16 \), so:
\[
x = 32
\]

### Step 5: Final calculation of \( GH \)
Now that we know \( x = 32 \), we substitute it into the formula for \( GH \):
\[
GH = \frac{15 \times 32}{16} = 30
\]

Thus, the length of \( GH \) is \( \boxed{30} \).

---

Would you like more details on any specific part, or do you have further questions?

Here are 5 questions to extend your understanding:
1. How would the distances change if the points were not collinear?
2. What if the distance \( DC \) was different, say 20? How would that affect \( GH \)?
3. How does the midpoint formula help in dividing line segments?
4. Can you generalize the formula for any two midpoints along a segment?
5. How would the result change if the coordinates were in 3D space?

**Tip:** For problems involving midpoints and distances, assigning coordinates can simplify the problem and avoid complex geometric constructions.

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

Solve a geometric problem involving midpoints of line segments and the distance between points. Learn how to find the length of GH when DC = 16 using midpoint and distance formulas. Suitable for Grades 9-12, this detailed solution covers key concepts in coordinate geometry.

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