https://file.aiquickdraw.com/mathbot/file/1722540636055mu5ovvs6.jpeg

To create the equations that represent the relationships in the bottom image, let's break down the different aspects shown in the table and graph:

### Information Extracted from the Bottom Image:
- **Height \( h \)**: 10 triangles
- **Total Triangles \( T \)**: 30 triangles
- **Perimeter \( P \)**: 20 toothpicks
- **Triangles with One Exposed Side**: 8 triangles
- **Triangles with Two Exposed Sides**: 6 triangles
- **Triangles with Three Exposed Sides**: 0 triangles
- **Triangles with No Exposed Sides**: 16 triangles

### Formulas:

1. **Height (h):**
   - \( h = 10 \)

2. **Total Triangles \( T \):**
   - \( T = \frac{h(h + 1)}{2} \)
   - For \( h = 10 \):
   \[
   T = \frac{10 \times 11}{2} = 55 \text{ triangles}
   \]
   - However, the image shows 30 triangles, suggesting that not all triangles are counted this way (possibly only considering the triangles forming full rows).

3. **Perimeter \( P \):**
   - Perimeter is measured in toothpicks, with each triangle side contributing to the perimeter if it is exposed.
   - The perimeter can be calculated by adding the sides of triangles on the edges.
   - For the grid configuration:
   \[
   P = 2h = 20 \text{ toothpicks}
   \]
   - The total number of exposed sides for the edge triangles is exactly equal to the perimeter.

4. **Triangles with Exposed Sides:**
   - **One Exposed Side:** These triangles are on the borders but not corners.
   \[
   \text{One Exposed Side} = 8
   \]
   - **Two Exposed Sides:** These are the corner triangles or those on the borders near the corners.
   \[
   \text{Two Exposed Sides} = 6
   \]
   - **Three Exposed Sides:** Typically only the smallest triangle at the top or corner would have all sides exposed, but it seems 0 in this configuration.
   \[
   \text{Three Exposed Sides} = 0
   \]
   - **No Exposed Sides:** Interior triangles, which are fully surrounded by other triangles.
   \[
   \text{No Exposed Sides} = 16
   \]

### Derived Equations for the Bottom Image:

1. **Total Triangles \( T \):**
   \[
   T = 30 \text{ triangles} 
   \text{ (as given in the table)}
   \]
   Since the traditional sum of rows formula doesn't match the observed value (which typically would give 55 for \( h = 10 \)), the actual count reflects a specific pattern for the grid shown.

2. **Perimeter \( P \):**
   \[
   P = 2h = 20 \text{ toothpicks}
   \]

3. **Number of Triangles with Exposed Sides:**
   - \( 8 + 6 + 0 + 16 = 30 \text{ (matches total triangles count)} \)

### Summary of Equations:

1. **Height \( h \)**: 10 triangles.
2. **Total Triangles \( T \)**: Given directly as 30.
3. **Perimeter \( P \)**: \( P = 2h = 20 \) toothpicks.
4. **Exposed Sides Breakdown:**
   \[
   \text{One Exposed Side} = 8, \quad \text{Two Exposed Sides} = 6, \quad \text{Three Exposed Sides} = 0, \quad \text{No Exposed Sides} = 16
   \]

These relationships summarize the patterns and match the given data.

Do you have any questions or need further clarification?

### 8 Related Questions:
1. How do these equations change if the grid is incomplete or irregular?
2. What would the equation for the total number of triangles look like if the arrangement is a different shape?
3. Can you derive a general formula for the perimeter for any height?
4. How do these patterns extend to 3D structures, like pyramids?
5. What if the triangles were equilateral and not right-angled? How would the equations change?
6. How does adding another row change the total number of triangles and the perimeter?
7. What happens to the perimeter calculation if the grid is extended infinitely?
8. Can you find a recursive relationship between perimeter and height?

### Tip:
When working with grid patterns, verify your equation by manually counting the elements for smaller sizes to ensure accuracy in the formula.

Learn how to derive equations for the number of triangles and perimeter in grid patterns with specific examples. This problem covers geometric concepts such as triangle numbers and perimeter calculations. Suitable for advanced high school students.

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