The problem essentially provides relationships between the sides of two triangles, $\Delta ABC$ and $\Delta EFG$, and states that the sides of $\Delta EFG$ are half the lengths of the corresponding sides of $\Delta ABC$.

Let’s break it down step by step:

Given Equalities:

1. $\frac{EF}{AB} = \frac{FG}{BC} = \frac{EG}{AC} = 2$.

This means that each side of triangle $\Delta ABC$ is twice as long as the corresponding side of triangle $\Delta EFG$, or equivalently:

$EF = \frac{1}{2} AB, \quad FG = \frac{1}{2} BC, \quad EG = \frac{1}{2} AC$

So, the sides of $\Delta EFG$ are exactly half the length of the corresponding sides of $\Delta ABC$.

Conclusion:

From the ratios given, we can conclude that $\Delta EFG$ is similar to $\Delta ABC$, with a similarity ratio of 1:2, meaning that the sides of $\Delta EFG$ are exactly half the lengths of the corresponding sides of $\Delta ABC$.

Would you like me to go into more details about the properties of similar triangles or how to derive further properties from this similarity?

Related Questions:

1. What are the properties of similar triangles?
2. How can the area of $\Delta EFG$ be compared to the area of $\Delta ABC$?
3. How do you prove two triangles are similar?
4. Can this problem be generalized to non-equilateral triangles?
5. What role does the ratio of side lengths play in triangle congruence?

Tip:

The ratio of corresponding side lengths in similar triangles is constant, but the ratio of their areas is the square of this ratio.