https://file.aiquickdraw.com/mathbot/file/17259740523051dozo48q.jpg

To address this problem, we need to determine whether it's possible to draw a polygon that is a scaled copy of both Polygon A and Polygon B, or prove that such a polygon cannot exist.

### Step 1: Analyze the Shapes

From the image:
- Polygon A and Polygon B appear similar in structure, but Polygon B is clearly larger than Polygon A.
- Both polygons have right angles and follow a stepped, horizontal arrangement, but their dimensions differ.

### Step 2: Check for Proportionality

A scaled copy of a polygon means that the lengths of the sides in one polygon are proportional to the lengths in the other polygon. To verify this, we need to check the relative proportions of the corresponding side lengths in both Polygon A and Polygon B.

Let's list the dimensions of the sides based on the grid:

- **Polygon A** appears to have side lengths:
  - Horizontal segments: 2 units, 3 units, 1 unit, and 2 units.
  - Vertical segments: 1 unit each.
  
- **Polygon B** appears to have side lengths:
  - Horizontal segments: 4 units, 6 units, 2 units, and 4 units.
  - Vertical segments: 2 units each.

### Step 3: Compare the Ratios

Now, let's compare the side lengths of the corresponding segments of both polygons:

- For the horizontal segments:
  - The ratio of the first horizontal segment (A: 2 units, B: 4 units) is 2:4 = 1:2.
  - The ratio of the second horizontal segment (A: 3 units, B: 6 units) is 3:6 = 1:2.
  - The ratio of the third horizontal segment (A: 1 unit, B: 2 units) is 1:2.
  - The ratio of the fourth horizontal segment (A: 2 units, B: 4 units) is 1:2.

- For the vertical segments:
  - The ratio of each vertical segment (A: 1 unit, B: 2 units) is 1:2.

### Step 4: Conclusion

Since all the corresponding side lengths of Polygon A and Polygon B have the same ratio of 1:2, it is possible to create a polygon that is a scaled copy of both Polygon A and Polygon B. Polygon B is simply a scaled-up version of Polygon A by a factor of 2.

Would you like more details on the scaling process or help with any further explanations? Here are some questions you might want to explore next:

1. How do you formally define a "scaled copy" in geometry?
2. What mathematical properties remain unchanged in a scaled polygon?
3. How can you calculate the area of a scaled polygon based on the original?
4. If a polygon is scaled by a factor of 2, what happens to its perimeter?
5. How does the concept of similarity apply to three-dimensional shapes?

**Tip:** When scaling a polygon, all corresponding angles remain the same, while the lengths of the sides change proportionally according to the scaling factor.

Is it possible to draw a polygon that is a scaled copy of both Polygon A and Polygon B? Either draw such a polygon, or explain how you know this is impossible.

Explore whether it's possible to draw a polygon that is a scaled version of both Polygon A and Polygon B. Learn about proportionality, scaling in geometry, and how to compare side lengths to determine if two shapes are scaled copies of each other. Suitable for middle school students.

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