To determine the length of $DC$, let's analyze the problem carefully:

Given:

1. $AB \parallel DC \parallel EF$ (parallel lines).
2. $E$ is the midpoint of $AC$, and $F$ lies on $BC$.
3. $EF = 10 \, \text{cm}$ and $AB = 55 \, \text{cm}$.
4. $AB, DC, EF$ are all parallel, so this forms a trapezoid where these lines decrease proportionally.

Key Observations:

Since $AB \parallel DC \parallel EF$, and $E$ is the midpoint of $AC$, the trapezoid has segments that decrease proportionally.

This can be solved using the concept of similar triangles or the Midline Theorem.

Midline Theorem:

The theorem states that in a triangle:

• A line segment connecting the midpoints of two sides is parallel to the third side and equals half its length.

Here, $EF$ connects midpoints of $AC$ and $BC$, so it is parallel to $AB$ and $DC$. Therefore: $EF = \frac{AB + DC}{2}$

We substitute the known values: $10 = \frac{55 + DC}{2}$

Solve for $DC$:

1. Multiply through by 2 to eliminate the denominator: $20 = 55 + DC$

2. Isolate $DC$: $DC = 20 - 55 = 30$

Final Answer:

The length of $DC$ is 30 cm.

Correct Option:

D. 30

Do you need further clarification on any step? Let me know!

Related Questions:

1. How does the Midline Theorem apply to trapezoids and triangles?
2. Can you explain how to find lengths in similar triangles?
3. How would the answer change if $EF$ were given as 15 cm instead?
4. What is the relationship between $AB$, $DC$, and $EF$ in proportional parallel lines?
5. How can coordinate geometry be used to solve a similar problem?

Tip:

When dealing with parallel lines and midpoints, always check for proportional relationships between segments. The Midline Theorem simplifies such problems significantly!