The perimeters of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle. 

Answer

To solve the problem of finding the corresponding side of the second triangle, we can follow these steps:

### Step 1: Understand the relationship between the triangles
Since the two triangles are similar, the ratio of their corresponding sides is equal to the ratio of their perimeters.

### Step 2: Write down the given information
- Perimeter of the first triangle (Triangle ABC) = 30 cm
- Perimeter of the second triangle (Triangle DEF) = 24 cm
- One side of the first triangle (AB) = 12 cm
- We need to find the corresponding side of the second triangle (DE).

### Step 3: Set up the ratio of the perimeters
The ratio of the perimeters of the two triangles can be expressed as:
\[ \frac{\text{Perimeter of Triangle ABC}}{\text{Perimeter of Triangle DEF}} = \frac{30}{24} \]

### Step 4: Simplify the ratio
To simplify the ratio:
\[ \frac{30}{24} = \frac{5}{4} \]
This means that the ratio of any corresponding sides of the triangles is also \( \frac{5}{4} \).

### Step 5: Set up the equation for the corresponding sides
Using the ratio of the sides, we can write:
\[ \frac{AB}{DE} = \frac{5}{4} \]
Substituting the known value of \( AB \):
\[ \frac{12}{DE} = \frac{5}{4} \]

### Step 6: Cross-multiply to solve for DE
Cross-multiplying gives:
\[ 12 \cdot 4 = 5 \cdot DE \]
\[ 48 = 5 \cdot DE \]

### Step 7: Solve for DE
Now, divide both sides by 5 to find DE:
\[ DE = \frac{48}{5} = 9.6 \text{ cm} \]

### Conclusion
The corresponding side of the second triangle (DE) is 9.6 cm.

---