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To prove the remaining rectangle is golden after removing a 1x1 square from a golden rectangle, follow these steps:

1. **Golden Mean Definition**: 
   The golden mean \( \tau = \frac{1 + \sqrt{5}}{2} \approx 1.618 \).
   
2. **Dimensions of Original Rectangle**: 
   The original rectangle has side lengths of 1 and \( \tau \) (since it's golden, the ratio of the longer side to the shorter side is \( \tau \)).

3. **Remove a 1x1 Square**: 
   Removing a 1x1 square leaves a smaller rectangle with dimensions \( (\tau - 1) \times 1 \).

4. **Check if the Remaining Rectangle is Golden**:
   The ratio of the remaining rectangle’s sides is:
   \[
   \frac{\tau - 1}{1} = \tau - 1
   \]
   Since \( \tau = \frac{1 + \sqrt{5}}{2} \), subtracting 1 gives:
   \[
   \tau - 1 = \frac{1 + \sqrt{5}}{2} - 1 = \frac{-1 + \sqrt{5}}{2} = \frac{\sqrt{5} - 1}{2}
   \]
   This is the inverse of \( \tau \), confirming the remaining rectangle has the same golden ratio, making it a golden rectangle.

Let me know if you'd like more details or further clarification.

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**Related Questions**:
1. What is the geometric significance of the golden ratio?
2. Can the golden ratio be found in nature or art?
3. How does the golden ratio relate to the Fibonacci sequence?
4. Why does subtracting a square from a golden rectangle result in another golden rectangle?
5. How can you calculate the golden ratio algebraically?

**Tip**: The golden ratio has many aesthetic and structural applications in both art and architecture.

Prove that the rectangle that remains after removing a 1x1 square from a golden rectangle is also a golden rectangle.

Explore how removing a 1x1 square from a golden rectangle results in another golden rectangle. This problem includes the definition of the golden mean and its application in geometric shapes, relevant to students in Grades 10-12.

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