If `DeltaPQR `and `DeltaLMN` are similar and `3PQ = LM and MN = 9 cm`, then QR is equal to :

Options

A. 12 cm
B. 6 cm
C. 9 cm
D. 3 cm

Answer

To solve the problem, we need to find the length of QR given that triangles ΔPQR and ΔLMN are similar, with the relationship \(3PQ = LM\) and \(MN = 9 \, \text{cm}\).

### Step-by-Step Solution:

1. **Understand the Similarity of Triangles**:
Since ΔPQR and ΔLMN are similar, the ratios of their corresponding sides are equal. This means:
\[ \frac{PQ}{LM} = \frac{QR}{MN} \]

2. **Express the Given Information**:
We know:
- \(3PQ = LM\)
- \(MN = 9 \, \text{cm}\)

3. **Substitute the Relationship**:
From \(3PQ = LM\), we can express \(LM\) in terms of \(PQ\):
\[ LM = 3PQ \]

4. **Set Up the Ratio**:
Now we can substitute \(LM\) into the ratio:
\[ \frac{PQ}{3PQ} = \frac{QR}{9} \]

5. **Simplify the Ratio**:
The left side simplifies to:
\[ \frac{1}{3} = \frac{QR}{9} \]

6. **Cross-Multiply to Solve for QR**:
Cross-multiplying gives us:
\[ QR = \frac{9}{3} = 3 \, \text{cm} \]

### Final Answer:
Thus, the length of QR is \(3 \, \text{cm}\).

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