To solve this problem, let’s analyze the triangle and the information provided:

Given:

1. $E$ is the midpoint of $AB$.
2. $G$ is the midpoint of $BC$.
3. We need to determine which of the statements about line segment $EG$ must be true.

Concept Used:

In geometry, a midsegment of a triangle connects the midpoints of two sides. Properties of the midsegment are:

1. The midsegment is parallel to the third side.
2. The length of the midsegment is half the length of the third side.

Here, $E$ and $G$ are midpoints of $AB$ and $BC$, respectively. Therefore, $EG$ is a midsegment of $\triangle ABC$.

Conclusion:

Using the properties of a midsegment:

1. $EG \parallel AC$ — The midsegment is parallel to the third side $AC$.
2. $EG = \frac{1}{2} \times AC$ — The length of $EG$ is half the length of $AC$.

Thus, the correct answer is: "EG \parallel AC".

Incorrect Options:

1. $EG$ is half the length of $BC$: Incorrect. $EG$ relates to $AC$, not $BC$.
2. $EG$ is half the length of $AB$: Incorrect. $EG$ does not relate directly to $AB$; it is parallel to $AC$.
3. $EG \perp AB$: Incorrect. $EG$ is not perpendicular to $AB$.

Would you like a detailed diagram explanation or further clarification?

Related Questions:

1. What is the midsegment theorem, and how is it derived?
2. How can you prove that $EG \parallel AC$ using coordinate geometry?
3. If $AB = 10$ and $BC = 12$, how would you find the length of $EG$?
4. Can you apply the midsegment theorem to other polygons like trapezoids?
5. How does the midsegment property help solve real-world geometry problems?

Tip:

The midsegment theorem is a fundamental property in triangles—always check for midpoints when determining parallelism and proportionality.