∆ABC ~ ∆DEF. If AB = 4 cm, BC = 3.5 cm, CA = 2.5 cm and DF = 7.5 cm, find the perimeter of ∆DEF

Answer

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

### Step 1: Calculate the perimeter of triangle ABC
The perimeter of a triangle is the sum of the lengths of its sides. For triangle ABC, we have:
- AB = 4 cm
- BC = 3.5 cm
- CA = 2.5 cm

So, the perimeter of triangle ABC (P_ABC) is:
\[ P_{ABC} = AB + BC + CA = 4 + 3.5 + 2.5 \]

### Step 2: Perform the addition
Now, let's add these values together:
\[ P_{ABC} = 4 + 3.5 + 2.5 = 10 \text{ cm} \]

### Step 3: Set up the proportion using the similarity of triangles
Since triangles ABC and DEF are similar, the ratios of their corresponding sides are equal. We can express this as:
\[ \frac{P_{ABC}}{P_{DEF}} = \frac{CA}{DF} \]
Where:
- CA = 2.5 cm
- DF = 7.5 cm

### Step 4: Substitute the known values into the proportion
Substituting the values we have:
\[ \frac{10}{P_{DEF}} = \frac{2.5}{7.5} \]

### Step 5: Simplify the ratio on the right side
The ratio \(\frac{2.5}{7.5}\) simplifies to:
\[ \frac{2.5}{7.5} = \frac{1}{3} \]

### Step 6: Set up the equation
Now we have:
\[ \frac{10}{P_{DEF}} = \frac{1}{3} \]

### Step 7: Cross-multiply to solve for \(P_{DEF}\)
Cross-multiplying gives us:
\[ 10 \cdot 3 = 1 \cdot P_{DEF} \]
\[ P_{DEF} = 30 \text{ cm} \]

### Conclusion
The perimeter of triangle DEF is 30 cm.

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