Similar Triangles: Finding Perimeter Using 9/100 Area Ratio

To find the perimeter of the small triangle, we need to follow these steps:

• Step 1: Understand the ratio of areas and relate it to the ratio of corresponding side lengths.
• Step 2: Use the areas' ratio to find the side length ratio.
• Step 3: Calculate the perimeter of the small triangle using the side length ratio.

First, recall the relationship between the areas of similar triangles and their side lengths: if two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Thus, we have:

$\left(\frac{\text{Side of small triangle}}{\text{Side of large triangle}}\right)^2 = \frac{\text{Area of small triangle}}{\text{Area of large triangle}} = \frac{9}{100}$

Taking the square root of both sides gives us the ratio of the side lengths:

$\frac{\text{Side of small triangle}}{\text{Side of large triangle}} = \sqrt{\frac{9}{100}} = \frac{3}{10}$

This tells us that each side of the small triangle is $\frac{3}{10}$ of the corresponding side of the large triangle. Consequently, this ratio applies to the perimeters of the triangles too.

Given that the perimeter of the large triangle is 129 cm, the perimeter of the small triangle is:

$\text{Perimeter of small triangle} = \frac{3}{10} \times 129 = 38.7 \, \text{cm}$

Therefore, the solution to the problem is $\mathbf{38.7}$ cm.