https://file.aiquickdraw.com/mathbot/file/1724896034389lmpmv32x.jpg

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Susan telah melukis corak segi tiga putih dan jingga. 

Empat corak yang pertama ditunjukkan seperti di bawah.  


Berapakah peratus segi tiga putih dalam Corak 4?   

The images provided show a sequence of triangular patterns, each subdivided into smaller triangles, and alternating between orange and white triangles. The sequence resembles the fractal pattern known as the Sierpinski triangle.

### Understanding the Pattern

1. **Pattern 1:** A single large orange triangle.
2. **Pattern 2:** The large triangle is subdivided into 4 smaller triangles, with the central triangle being white.
3. **Pattern 3:** Each of the smaller orange triangles from Pattern 2 is further subdivided into 4 smaller triangles, again leaving the central triangle of each as white.
4. **Pattern 4:** This pattern follows the same rule as Pattern 3, where each orange triangle from Pattern 3 is further subdivided, making even more white triangles.

### Calculation of White Triangles in Pattern 4

For each pattern step, the number of smaller white triangles increases.

- **Level 0 (Pattern 1):** 1 large orange triangle, 0 white triangles.
- **Level 1 (Pattern 2):** 4 smaller triangles (1 large orange with 1 white in the middle, 3 smaller orange). This creates 1 white triangle.
- **Level 2 (Pattern 3):** Each of the 3 orange triangles from Level 1 subdivides into 4, adding 3 more white triangles, totaling 1 (from Level 1) + 3 = 4 white triangles.
- **Level 3 (Pattern 4):** Each of the 9 orange triangles from Level 2 subdivides into 4, adding 9 more white triangles, totaling 4 (from Level 2) + 9 = 13 white triangles.

The general formula for the number of white triangles at any level \(n\) (where \(n = 0, 1, 2, 3, \ldots\)) is given by:
\[
\text{Number of white triangles} = \frac{3^n - 1}{2}
\]

### Number of Total Triangles

At Level \(n\), the total number of smaller triangles formed is:
\[
\text{Total triangles} = 3^n
\]

For **Pattern 4 (n = 4):**

- **Total triangles** = \(3^4 = 81\)
- **White triangles** = \(\frac{3^4 - 1}{2} = \frac{81 - 1}{2} = 40\)

### Percentage of White Triangles

To calculate the percentage of white triangles in Pattern 4:

\[
\text{Percentage of white triangles} = \left( \frac{\text{Number of white triangles}}{\text{Total triangles}} \right) \times 100
\]

Substitute the values:

\[
\text{Percentage of white triangles} = \left( \frac{40}{81} \right) \times 100 \approx 49.38\%
\]

### Conclusion

The percentage of white triangles in Pattern 4 is approximately **49.38%**.

Would you like a more detailed explanation or have any specific questions about the calculations?

### Relative Questions for Further Exploration:

1. How does the percentage of white triangles change as the pattern continues to higher levels?
2. What is the relationship between the number of orange triangles and the total number of triangles in any given pattern level?
3. Can you derive a formula for the number of orange triangles at any given pattern level?
4. How does the concept of fractals apply to this triangular pattern?
5. What would be the percentage of white triangles at Pattern 5?

### Tip:

When working with fractal patterns like the Sierpinski triangle, recognizing the self-similar nature can help in deriving formulas for geometric and arithmetic properties across levels.

Learn how to calculate the percentage of white triangles in Pattern 4 of the Sierpinski triangle. Detailed step-by-step solution including the fractal geometry concept and formula derivation. Suitable for advanced high school students and fractal geometry enthusiasts.

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