In `Delta` PQR, PQ = 4 cm, QR = 3 cm, and RP = 3.5 cm. `Delta` DEF is similar to `Delta` PQR. If EF = 9 cm, then what is the perimeter of `Delta` DEF ?

Options

A. 10.5 cm
B. 21 cm
C. 31.5 cm
D. Cannot be determined as data is insufficient

Answer

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

### Step 1: Understand the similarity of triangles
Since triangles DEF and PQR are similar, the ratios of their corresponding sides are equal. This means:
\[ \frac{PQ}{DE} = \frac{QR}{EF} = \frac{RP}{DF} \]

### Step 2: Write down the known values
From the question, we have:
- \(PQ = 4 \, \text{cm}\)
- \(QR = 3 \, \text{cm}\)
- \(RP = 3.5 \, \text{cm}\)
- \(EF = 9 \, \text{cm}\)

### Step 3: Find the length of side DE
Using the ratio of sides:
\[ \frac{PQ}{DE} = \frac{QR}{EF} \]
Substituting the known values:
\[ \frac{4}{DE} = \frac{3}{9} \]
Cross-multiplying gives:
\[ 4 \times 9 = 3 \times DE \]
\[ 36 = 3 \times DE \]
Now, divide both sides by 3:
\[ DE = \frac{36}{3} = 12 \, \text{cm} \]

### Step 4: Find the length of side DF
Using the ratio of sides again:
\[ \frac{RP}{DF} = \frac{QR}{EF} \]
Substituting the known values:
\[ \frac{3.5}{DF} = \frac{3}{9} \]
Cross-multiplying gives:
\[ 3.5 \times 9 = 3 \times DF \]
\[ 31.5 = 3 \times DF \]
Now, divide both sides by 3:
\[ DF = \frac{31.5}{3} = 10.5 \, \text{cm} \]

### Step 5: Calculate the perimeter of triangle DEF
The perimeter \(P\) of triangle DEF is the sum of its sides:
\[ P = DE + EF + DF \]
Substituting the values we found:
\[ P = 12 + 9 + 10.5 = 31.5 \, \text{cm} \]

### Final Answer
The perimeter of triangle DEF is \(31.5 \, \text{cm}\).

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