If `DeltaABC ~ DeltaDEF` AB =4 cm , DE = 6 cm, EF =9 cm and FD =12 cm find the perimeter of `DeltaABC ` .

Answer

To find the perimeter of triangle ABC given that triangle ABC 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 their corresponding sides are equal.

### Step 2: Set up the ratios
We know:
- AB = 4 cm
- DE = 6 cm
- EF = 9 cm
- FD = 12 cm

From the similarity, we can set up the following ratios:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{FD} \]

### Step 3: Calculate BC using the first ratio
Using the first ratio:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Substituting the known values:
\[ \frac{4}{6} = \frac{BC}{9} \]
Cross-multiplying gives:
\[ 4 \times 9 = 6 \times BC \]
\[ 36 = 6 \times BC \]
Dividing both sides by 6:
\[ BC = \frac{36}{6} = 6 \text{ cm} \]

### Step 4: Calculate AC using the second ratio
Using the second ratio:
\[ \frac{AB}{DE} = \frac{AC}{FD} \]
Substituting the known values:
\[ \frac{4}{6} = \frac{AC}{12} \]
Cross-multiplying gives:
\[ 4 \times 12 = 6 \times AC \]
\[ 48 = 6 \times AC \]
Dividing both sides by 6:
\[ AC = \frac{48}{6} = 8 \text{ cm} \]

### Step 5: Calculate the perimeter of triangle ABC
The perimeter of triangle ABC is the sum of its sides:
\[ \text{Perimeter} = AB + BC + AC \]
Substituting the values we found:
\[ \text{Perimeter} = 4 + 6 + 8 = 18 \text{ cm} \]

### Final Answer
The perimeter of triangle ABC is **18 cm**.

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