`triangle ABC - triangle DEF` such that ` DE = 3 cm, EF = 2 cm, DF = 2.5 cm " and " BC = 4 cm.` Find perimeter of `triangle ABC`.

Answer

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

### Step 1: Understand the Similarity of Triangles
Since triangle ABC is similar to triangle DEF, the ratios of the lengths of corresponding sides are equal.

### Step 2: Identify the Given Values
From the problem, we have:
- DE = 3 cm
- EF = 2 cm
- DF = 2.5 cm
- BC = 4 cm

### Step 3: Set Up the Ratios
Using the similarity of the triangles, we can set up the following ratios for the corresponding sides:
- AB / DE = BC / EF
- AC / DF = BC / EF

### Step 4: Calculate AB
Using the ratio of sides:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Substituting the known values:
\[ \frac{AB}{3} = \frac{4}{2} \]
This simplifies to:
\[ \frac{AB}{3} = 2 \]
Now, cross-multiply to find AB:
\[ AB = 2 \times 3 = 6 \text{ cm} \]

### Step 5: Calculate AC
Now, we can find AC using the ratio:
\[ \frac{AC}{DF} = \frac{BC}{EF} \]
Substituting the known values:
\[ \frac{AC}{2.5} = \frac{4}{2} \]
This simplifies to:
\[ \frac{AC}{2.5} = 2 \]
Cross-multiply to find AC:
\[ AC = 2 \times 2.5 = 5 \text{ cm} \]

### Step 6: Calculate the Perimeter of Triangle ABC
Now that we have all the sides of triangle ABC:
- AB = 6 cm
- BC = 4 cm
- AC = 5 cm

We can find the perimeter:
\[ \text{Perimeter} = AB + BC + AC \]
Substituting the values:
\[ \text{Perimeter} = 6 + 4 + 5 = 15 \text{ cm} \]

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

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