https://file.aiquickdraw.com/mathbot/file/172674668250239qoyer8.jpg

The problem involves a triangle with minions posing inside it, and it mentions a scale factor of 3. The task is to fill in the chart and write specific ratios.

Let's break down the image's components and calculations.

### PART A: Fill in the chart
Given dimensions:
- The original triangle has sides of **3**, **2**, and **3.7** units (triangle with minions inside).
- The new image is scaled by a factor of **3**.

Using the scale factor, the new dimensions will be:

1. **Dimensions of the new image**:
   - New side 1: \( 3 \times 3 = 9 \)
   - New side 2: \( 2 \times 3 = 6 \)
   - New side 3: \( 3.7 \times 3 = 11.1 \)

2. **Perimeter**:
   - Original perimeter = \( 3 + 2 + 3.7 = 8.7 \)
   - New perimeter = \( 9 + 6 + 11.1 = 26.1 \)

3. **Area**:
   The area of a triangle is given by:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Assuming the base is **2** and the height is **3**, the original area would be:
   \[
   \text{Original area} = \frac{1}{2} \times 2 \times 3 = 3 \, \text{square units}
   \]
   The new area, using the scale factor (area scales by the square of the scale factor):
   \[
   \text{New area} = 3 \times (3^2) = 3 \times 9 = 27 \, \text{square units}
   \]

### PART B: Write the following ratios
- **Model length: Actual length** = \( \frac{1}{3} \)
- **Model perimeter: Actual perimeter** = \( \frac{8.7}{26.1} = \frac{1}{3} \)
- **Model area: Actual area** = \( \frac{3}{27} = \frac{1}{9} \)

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Would you like any further details or explanations?

### Additional Questions:
1. How would the dimensions change with a scale factor of 4 instead of 3?
2. What if the triangle was not equilateral but had different side lengths?
3. How does the scale factor impact the volume of a 3D shape if the shape was a pyramid?
4. Could you calculate the area using Heron’s formula instead of base-height?
5. What is the relationship between perimeter ratios and area ratios in general?

#### Tip:
For any shape, area scales by the square of the scale factor, while perimeter scales linearly.

The problem involves a triangle with minions posing inside it, and it mentions a scale factor of 3. The task is to fill in the chart and write specific ratios.

In this problem, learn how to scale a triangle by a factor of 3, involving the calculation of new dimensions, perimeter, and area. Understand key concepts such as the linear scaling of perimeter and the quadratic scaling of area. Suitable for middle school students learning geometry and ratios.

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