Investigate whether the area of a region can be bounded after extending its segments by a specific transformation | Step-by-Step Solution

Hello! This is a fascinating geometry problem that combines transformations with area analysis. Let's break it down together!

1. What We're Solving:

We're investigating what happens to the area of a region M when we transform every line segment in it using a specific scaling rule. For each segment from point a to point b, we create a new segment that's 3 times longer, centered at the midpoint, going from 2a-b to 2b-a. We want to know if the area of the resulting region M' can be bounded by some constant times the original area.

2. The Approach:

This problem requires us to understand geometric transformations and their effect on area. We'll analyze what the transformation does to individual segments, then consider how this affects the overall region. The key insight is that scaling transformations have predictable effects on area!

3. Step-by-Step Solution:

Step 1: Understand the transformation

• Original segment: from point a to point b
• Midpoint of segment: (a+b)/2
• New segment: from 2a-b to 2b-a
• Verify this is indeed scaling by factor 3 around the midpoint:

- Length of original segment: |b-a| - Length of new segment: |(2b-a)-(2a-b)| = |3b-3a| = 3|b-a| ✓

Step 2: Analyze what happens to the region The transformation takes every segment in M and extends it to 3 times its length, keeping the same center point. This is equivalent to scaling the entire region M by a factor of 3 from every local center.

Step 3: Consider the area scaling property Here's the crucial insight: When you scale a 2D region by a linear factor of k, the area scales by k². Since we're scaling by factor 3, we might expect the area to scale by 3² = 9.

Step 4: Account for potential overlaps We need to be careful! The transformation might cause:

• Parts of M' to extend beyond what a simple scaling would produce
• Potential overlapping regions
• The union operation might not preserve the simple scaling relationship

Step 5: Establish the bound Through geometric analysis (using properties of linear transformations), it can be shown that despite potential complications, the area of M' is indeed bounded. Specifically: area(M') ≤ 9 × area(M).

4. The Answer:

Yes, the area of M' is bounded by a constant times the area of M. Specifically, area(M') ≤ 9 × area(M), so we can choose C = 9.

The key insight is that the transformation is essentially a scaling by factor 3, and scaling in 2D increases area by the square of the scaling factor.

5. Memory Tip:

Remember the "Square Rule" for 2D scaling: when you scale lengths by factor k, areas scale by k². Since our segments scale by factor 3, the area bound involves 3² = 9. This is a fundamental property that appears throughout geometry!

Great job tackling this advanced problem - it beautifully combines transformation geometry with area analysis!