
The problem states that:

A
B
E
F
=
B
C
F
G
=
A
C
E
G
=
2.
EF
AB
​
 = 
FG
BC
​
 = 
EG
AC
​
 =2.
Thus, the sides of ΔEFG are exactly half the lengths of the corresponding sides of ΔABC.

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.

The problem states that: A B E F = B C F G = A C E G = 2. Thus, the sides of ΔEFG are exactly half the lengths of the corresponding sides of ΔABC.

This problem explores the similarity between triangles ΔEFG and ΔABC, where the sides of ΔEFG are half the lengths of the corresponding sides of ΔABC. Learn how to apply triangle similarity theorems and ratio properties in geometry. Suitable for students in Grades 8-10.

Math Problem Statement

A B E F

B C F G

A C E G

EF AB

FG BC

Solution

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