`DeltaABC` is such that AB = `3` cm, BC = `2` cm, CA = `2.5` cm. If `DeltaABC`~`DeltaDEF` and EF= `4` cm, then perimeter of `DeltaDEF` is :

Options

A. `7.5` cm
B. `15` cm
C. `22.5` cm
D. `30` cm

Answer

To solve the problem step by step, we will use the properties of similar triangles.

### Step 1: Identify the sides of triangle ABC
Given:
- AB = 3 cm
- BC = 2 cm
- CA = 2.5 cm

### Step 2: Identify the side EF of triangle DEF
Given:
- EF = 4 cm

### Step 3: Set up the ratio of the sides of the similar triangles
Since triangle ABC is similar to triangle DEF, we can set up the following ratios:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF} \]

### Step 4: Calculate DE using the ratio of AB and EF
From the ratio:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Substituting the known values:
\[ \frac{3}{DE} = \frac{2}{4} \]
Cross-multiplying gives:
\[ 3 \cdot 4 = 2 \cdot DE \]
\[ 12 = 2 \cdot DE \]
Now, divide both sides by 2:
\[ DE = \frac{12}{2} = 6 \text{ cm} \]

### Step 5: Calculate DF using the ratio of BC and EF
Now, we use the ratio:
\[ \frac{BC}{EF} = \frac{CA}{DF} \]
Substituting the known values:
\[ \frac{2}{4} = \frac{2.5}{DF} \]
Cross-multiplying gives:
\[ 2 \cdot DF = 4 \cdot 2.5 \]
Calculating the right side:
\[ 2 \cdot DF = 10 \]
Now, divide both sides by 2:
\[ DF = \frac{10}{2} = 5 \text{ cm} \]

### Step 6: Calculate the perimeter of triangle DEF
The perimeter of triangle DEF is given by:
\[ \text{Perimeter} = DE + EF + DF \]
Substituting the values we found:
\[ \text{Perimeter} = 6 + 4 + 5 \]
Calculating the sum:
\[ \text{Perimeter} = 15 \text{ cm} \]

### Final Answer
The perimeter of triangle DEF is **15 cm**.

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