https://file.aiquickdraw.com/mathbot/file/1727392360105tnhxu7tq.jpg

Find the ratio of the perimeters.  That is, find perimeter new/perimeter original.  What do you notice?

the first "U" is 6 units width and 8 units length, and the second "U" is 9 units width and 12 units length.

Let's solve the problem based on the image provided and the details mentioned. 

We are tasked with finding the ratio of the perimeters between two "U"-shaped figures.

### Dimensions:
1. **Original "U" Shape:**
   - Width: 6 units
   - Height: 8 units

2. **New "U" Shape:**
   - Width: 9 units
   - Height: 12 units

### Step 1: Perimeter of the original "U"-shaped figure

- The perimeter of the original shape includes the outer perimeter and the inner perimeter (subtracting the "cut-out").
- Outer perimeter: \( 2(\text{height} + \text{width}) = 2(8 + 6) = 28 \) units.
- Inner perimeter (cut-out part): \( 2(\text{inner height} + \text{inner width}) = 2((8 - 2) + (6 - 2)) = 2(6 + 4) = 20 \) units.
  
So, the total perimeter of the original figure is:
\[
P_{\text{original}} = 28 + 20 = 48 \text{ units}.
\]

### Step 2: Perimeter of the new "U"-shaped figure

Similarly, the new "U"-shaped figure has:
- Outer perimeter: \( 2(\text{height} + \text{width}) = 2(12 + 9) = 42 \) units.
- Inner perimeter (cut-out part): \( 2(\text{inner height} + \text{inner width}) = 2((12 - 3) + (9 - 3)) = 2(9 + 6) = 30 \) units.
  
So, the total perimeter of the new figure is:
\[
P_{\text{new}} = 42 + 30 = 72 \text{ units}.
\]

### Step 3: Ratio of the perimeters

Now, let's find the ratio of the perimeters:
\[
\text{Ratio} = \frac{P_{\text{new}}}{P_{\text{original}}} = \frac{72}{48} = \frac{3}{2}.
\]

### Conclusion:

The ratio of the perimeters is \( \frac{3}{2} \).

### Observation:
The ratio of the perimeters corresponds directly to the ratio of the dimensions (both height and width) of the "U" shapes.

---

Would you like more details or explanations? Here are some additional questions you can explore:

1. How does the area ratio compare to the perimeter ratio in this case?
2. What happens to the perimeter if we scale both dimensions by a different factor?
3. Can you derive a general formula for the perimeter of a "U"-shaped figure like this?
4. How does the shape's complexity affect the relationship between perimeter and area?
5. How would the ratio change if the inner cut-out part scaled differently?

**Tip:** Always double-check calculations involving multiple parts of a shape to ensure all components are accounted for correctly.

Find the ratio of the perimeters. That is, find perimeter new/perimeter original. What do you notice? The first 'U' is 6 units width and 8 units length, and the second 'U' is 9 units width and 12 units length.

Explore the problem of finding the ratio of the perimeters of two U-shaped figures with different dimensions. The solution involves using perimeter formulas and understanding how scaling affects the perimeter. Ideal for students in Grades 6-8.

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