`DeltaABC ~DeltaDEF`. If AB= 4cm,BC = 3.5cm,CA = 2.5cmandDF = 7.5cm, then find perimeter of `DeltaDEF`.

Answer

To find the perimeter of triangle DEF given that triangle ABC is similar to triangle DEF, we can follow these steps:

### Step 1: Write down the similarity ratio
Since triangles ABC and DEF are similar, we can write the ratios of their corresponding sides:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF} \]

### Step 2: Substitute the known values
We know:
- \( AB = 4 \, \text{cm} \)
- \( BC = 3.5 \, \text{cm} \)
- \( CA = 2.5 \, \text{cm} \)
- \( DF = 7.5 \, \text{cm} \)

Using the ratio \( \frac{BC}{EF} = \frac{CA}{DF} \):
\[ \frac{3.5}{EF} = \frac{2.5}{7.5} \]

### Step 3: Cross-multiply to find EF
Cross-multiplying gives us:
\[ 3.5 \times 7.5 = 2.5 \times EF \]
Calculating \( 3.5 \times 7.5 \):
\[ 3.5 \times 7.5 = 26.25 \]
So, we have:
\[ 26.25 = 2.5 \times EF \]

### Step 4: Solve for EF
Now, divide both sides by 2.5:
\[ EF = \frac{26.25}{2.5} = 10.5 \, \text{cm} \]

### Step 5: Use the ratio \( \frac{AB}{DE} = \frac{CA}{DF} \) to find DE
Now we use the ratio:
\[ \frac{AB}{DE} = \frac{CA}{DF} \]
Substituting the known values:
\[ \frac{4}{DE} = \frac{2.5}{7.5} \]

### Step 6: Cross-multiply to find DE
Cross-multiplying gives us:
\[ 4 \times 7.5 = 2.5 \times DE \]
Calculating \( 4 \times 7.5 \):
\[ 4 \times 7.5 = 30 \]
So, we have:
\[ 30 = 2.5 \times DE \]

### Step 7: Solve for DE
Now, divide both sides by 2.5:
\[ DE = \frac{30}{2.5} = 12 \, \text{cm} \]

### Step 8: Find the perimeter of triangle DEF
Now that we have all the sides:
- \( DE = 12 \, \text{cm} \)
- \( EF = 10.5 \, \text{cm} \)
- \( DF = 7.5 \, \text{cm} \)

The perimeter \( P \) of triangle DEF is:
\[ P = DE + EF + DF = 12 + 10.5 + 7.5 \]
Calculating the sum:
\[ P = 30 \, \text{cm} \]

### Final Answer
The perimeter of triangle DEF is \( 30 \, \text{cm} \).