https://file.aiquickdraw.com/mathbot/file/1727775574121sqdrf13g.jpg

This problem is focused on geometric relationships within a regular pentagon \( ABCDE \) with side length 1. Here's a breakdown of the questions from the image:

1. **(1) Prove that \( FG = 2 - 2x \):** 
   - Points \( F \), \( G \) are intersections between diagonals \( AC \), \( BE \), and \( AD \), \( BE \), respectively. The challenge is to demonstrate that the length of \( FG \) can be expressed as \( 2 - 2x \), where \( x \) represents the length of diagonal \( AC \).

2. **(2) Find the value of \( x \):**
   - Using geometric properties of a regular pentagon, the task is to calculate the value of \( x \), the length of diagonal \( AC \).

3. **(3) Express vector \( \overrightarrow{AC} \) using vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AE} \):**
   - The problem asks to rewrite \( \overrightarrow{AC} \) in terms of the two vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AE} \), which would likely involve decomposing it into vector components.

Would you like help with solving any specific part, or shall I proceed with working through all three parts in detail?

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Here are five relative questions for deeper exploration:

1. How can you calculate the intersection points of diagonals in a regular pentagon?
2. What geometric properties of regular pentagons help in proving relationships between diagonals and sides?
3. Can vector decomposition be used for shapes other than regular polygons, and if so, how?
4. How does symmetry in a pentagon simplify the calculation of distances between intersection points?
5. Why do regular polygons often appear in problems involving vector components?

**Tip:** Regular polygons, like pentagons, have rich symmetry, which often simplifies geometric and algebraic solutions.

In a regular pentagon ABCDE with side length 1, diagonal AC and BE intersect at point F, and AD and BE intersect at point G. Given AC = x, solve the following: (1) Prove that FG = 2 - 2x. (2) Find the value of x. (3) Express vector AC in terms of AB and AE.

Solve a geometric problem involving a regular pentagon ABCDE, where the challenge is to prove the relationship between intersection points of diagonals, calculate the value of a diagonal, and express vector AC in terms of other vectors. Suitable for Grades 11-12.

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