`Delta ABC~Delta DEF` and their perimeters are 32 cm and 24 cm respectively. If AB=10 cm then DE=?

Options

A. `8` cm
B. `7.5` cm
C. `15` cm
D. `5sqrt(3)` cm

Answer

To solve the problem, we will use the concept of similar triangles and the relationship between their corresponding sides and perimeters.

### Step-by-Step Solution:

1. **Identify the given information:**
- The perimeters of triangles ABC and DEF are given as:
- Perimeter of ΔABC = 32 cm
- Perimeter of ΔDEF = 24 cm
- The length of side AB is given as:
- AB = 10 cm

2. **Set up the ratio of the perimeters:**
Since the triangles are similar (ΔABC ~ ΔDEF), the ratio of their corresponding sides is equal to the ratio of their perimeters. Therefore, we can set up the following ratio:
\[ \frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta DEF} = \frac{AB}{DE} \]
Substituting the known values:
\[ \frac{32}{24} = \frac{10}{DE} \]

3. **Simplify the ratio of the perimeters:**
We can simplify \(\frac{32}{24}\):
\[ \frac{32}{24} = \frac{4}{3} \]
So we have:
\[ \frac{4}{3} = \frac{10}{DE} \]

4. **Cross-multiply to find DE:**
Cross-multiplying gives us:
\[ 4 \cdot DE = 3 \cdot 10 \]
Simplifying this:
\[ 4 \cdot DE = 30 \]

5. **Solve for DE:**
Now, divide both sides by 4 to isolate DE:
\[ DE = \frac{30}{4} = 7.5 \text{ cm} \]

### Final Answer:
\[ DE = 7.5 \text{ cm} \]