Certainly! Let's solve this step-by-step.
### Step 1: Determine the Lengths of the Sides of the First Triangle
The sides of the first triangle are in the ratio [tex]\(3:4:5\)[/tex]. We are also given that the perimeter of this triangle is [tex]\(96 \, \text{cm}\)[/tex].
Let's denote the sides as [tex]\(3x\)[/tex], [tex]\(4x\)[/tex], and [tex]\(5x\)[/tex]. The perimeter equation will be:
[tex]\[
3x + 4x + 5x = 96
\][/tex]
Combine the terms:
[tex]\[
12x = 96
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{96}{12} = 8
\][/tex]
Now, the lengths of the sides of the first triangle can be calculated:
[tex]\[
3x = 3 \times 8 = 24 \, \text{cm}
\][/tex]
[tex]\[
4x = 4 \times 8 = 32 \, \text{cm}
\][/tex]
[tex]\[
5x = 5 \times 8 = 40 \, \text{cm}
\][/tex]
### Step 2: Determine the Scale Factor Between the Two Triangles
We are told that the longest side of a similar triangle is [tex]\(20 \, \text{cm}\)[/tex], and the longest side of the first triangle is [tex]\(40 \, \text{cm}\)[/tex].
The scale factor (k) between the two similar triangles is:
[tex]\[
k = \frac{\text{Length of a corresponding side of the similar triangle}}{\text{Length of a corresponding side of the first triangle}} = \frac{20}{40} = 0.5
\][/tex]
### Step 3: Determine the Ratio of Their Areas
The ratio of the areas of two similar triangles is the square of the scale factor between corresponding sides.
Thus, the ratio of the areas is:
[tex]\[
\text{Area ratio} = k^2 = (0.5)^2 = 0.25
\][/tex]
### Conclusion
The ratio of the areas of the two similar triangles is:
[tex]\[
\boxed{0.25}
\][/tex]
