Let's break down the problem based on the provided image.

Problem Summary:

1. Figure 1 is an equilateral triangle with a total area of 100 cm².
2. In Figure 2, a smaller equilateral triangle (created by joining the midpoints of the original triangle) is removed, leaving three smaller triangles.
3. This process is repeated for Figure 3, Figure 4, and so on, creating a fractal-like pattern.
4. The shaded area of each figure forms a geometric sequence.

We need to address the following:

Part (a): Write down the common ratio, $r$.

• In Figure 1, the area is 100 cm².
• In Figure 2, a smaller triangle (1/4 of the original area) is removed, leaving 3/4 of the area shaded.
  • So, the shaded area in Figure 2 is $100 \times \frac{3}{4} = 75 \, \text{cm}^2$.

Thus, the common ratio $r$ is $\frac{3}{4}$.

Part (b): Find the shaded area in Figure 10.

• The shaded area follows a geometric sequence with:
  • Initial area $a = 100 \, \text{cm}^2$
  • Common ratio $r = \frac{3}{4}$

The formula for the $n$-th term in a geometric sequence is: $A_n = A_1 \times r^{n-1}$ For Figure 10 ($n = 10$): $A_{10} = 100 \times \left(\frac{3}{4}\right)^{9}$

Let's calculate this value.

Part (c): Find the total shaded areas of Figures 1 to 10.

• The sum of the first $n$ terms of a geometric sequence is given by: $S_n = \frac{a(1 - r^n)}{1 - r}$ For Figures 1 to 10 ($n = 10$, $a = 100$, and $r = \frac{3}{4}$): $S_{10} = \frac{100(1 - \left(\frac{3}{4}\right)^{10})}{1 - \frac{3}{4}}$

I'll calculate both (b) and (c) now.### Results:

• (b) The shaded area in Figure 10 is approximately 7.51 cm².
• (c) The total shaded area of Figures 1 to 10 is approximately 377.47 cm².

Would you like further details or clarifications on any of these calculations?

Here are 5 related questions to expand on this topic:

1. How does the area reduction pattern change if a different fraction of the triangle is removed?
2. What would happen to the total shaded area if the process continued indefinitely?
3. Can the total area ever become zero? Why or why not?
4. How does this fractal pattern relate to other famous fractals like the Sierpinski triangle?
5. How would the solution change if the initial triangle's area were different?

Tip: When working with geometric sequences, always ensure you carefully identify the initial term and common ratio before applying formulas.