To find the area of the smaller triangle, we can use the property of similar figures. The areas of similar figures are proportional to the square of the ratio of their corresponding sides.

Given:

• The area of the larger triangle = 195 ft².
• The length of the corresponding side of the larger triangle = 12 ft.
• The length of the corresponding side of the smaller triangle = 9 ft.

Step 1: Find the ratio of the corresponding sides

The ratio of the sides of the smaller triangle to the larger triangle is: $\text{Ratio of sides} = \frac{\text{Smaller side}}{\text{Larger side}} = \frac{9}{12} = \frac{3}{4}$

Step 2: Use the square of the ratio to find the ratio of areas

The ratio of the areas of two similar figures is the square of the ratio of their corresponding sides. Thus, the ratio of the areas is: $\text{Ratio of areas} = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$

Step 3: Set up the proportion to find the area of the smaller triangle

Let $A_{\text{smaller}}$ be the area of the smaller triangle. We can set up the proportion: $\frac{A_{\text{smaller}}}{A_{\text{larger}}} = \frac{9}{16}$

Substitute the given area of the larger triangle: $\frac{A_{\text{smaller}}}{195} = \frac{9}{16}$

Now, solve for $A_{\text{smaller}}$: $A_{\text{smaller}} = 195 \times \frac{9}{16} = 195 \times 0.5625 = 109.6875 \, \text{ft}^2$

Step 4: Round to the nearest whole number

The area of the smaller triangle is approximately: $A_{\text{smaller}} \approx 110 \, \text{ft}^2$

So, the area of the smaller triangle is 110 ft².

---

Would you like more details on any part of the solution? Or have any other questions?

Here are some related questions for further practice:

1. What happens if the ratio of corresponding sides changes to $\frac{6}{8}$?
2. How does the area of the smaller triangle change if the area of the larger triangle were doubled?
3. What is the perimeter ratio of two similar triangles if the side lengths have a ratio of $\frac{5}{8}$?
4. If the area of the larger triangle were 400 ft², how would that change the area of the smaller triangle?
5. How can you use the proportionality of similar figures to solve for unknown side lengths when the areas are given?

Tip: When solving problems involving similar figures, always remember to use the ratio of corresponding sides squared when dealing with areas!