In `DeltaPQR`, PQ = 4 cm, QR = 3 cm and RP = 3.5 cm, `DeltaDEF` is similar to `DeltaPQR`. If EF = 9 cm then perimeter of `DeltaDEF` is -

Options

A. 10.5 cm
B. 21 cm
C. 31.5 cm
D. Cannot be determined

Answer

To find the perimeter of triangle DEF, we will use the properties of similar triangles. Here are the steps to solve the problem:

### Step 1: Find the perimeter of triangle PQR
The perimeter of triangle PQR is the sum of its sides. Given:
- PQ = 4 cm
- QR = 3 cm
- RP = 3.5 cm

**Calculation:**
\[ \text{Perimeter of } \Delta PQR = PQ + QR + RP = 4 + 3 + 3.5 = 10.5 \text{ cm} \]

### Step 2: Set up the ratio of the perimeters
Since triangles DEF and PQR are similar, the ratio of their perimeters is equal to the ratio of their corresponding sides. We can express this as:
\[ \frac{\text{Perimeter of } \Delta DEF}{\text{Perimeter of } \Delta PQR} = \frac{EF}{QR} \]

### Step 3: Substitute the known values
We know:
- EF = 9 cm
- QR = 3 cm
- Perimeter of PQR = 10.5 cm

Substituting these values into the ratio gives:
\[ \frac{\text{Perimeter of } \Delta DEF}{10.5} = \frac{9}{3} \]

### Step 4: Simplify the ratio
Simplifying the right side:
\[ \frac{9}{3} = 3 \]

### Step 5: Solve for the perimeter of triangle DEF
Now we can solve for the perimeter of triangle DEF:
\[ \text{Perimeter of } \Delta DEF = 10.5 \times 3 = 31.5 \text{ cm} \]

### Final Answer:
The perimeter of triangle DEF is **31.5 cm**.

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