If `DeltaABC~DeltaPQR` with `(BC)/(QR)=1/3`,then `(ar(DeltaPRQ))/(ar(DeltaBCA))` is equal to

Options

A. `9/1`
B. `3/1`
C. `1/3`
D. `1/9`

Explanation

To solve the problem, we need to find the ratio of the areas of triangles PRQ and BCA given that triangle ABC is similar to triangle PQR and that the ratio of the sides BC and QR is 1/3.

### Step-by-Step Solution:

1. **Understand the Similarity of Triangles**:
Since ΔABC ~ ΔPQR, we know that the corresponding sides of similar triangles are in proportion.

2. **Identify the Given Ratio**:
We are given that \(\frac{BC}{QR} = \frac{1}{3}\).

3. **Find the Ratio of QR to BC**:
From the given ratio, we can find the inverse ratio:
\[ \frac{QR}{BC} = \frac{3}{1} \]

4. **Use the Area Ratio Theorem**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{ar(ΔPRQ)}{ar(ΔBCA)} = \left(\frac{QR}{BC}\right)^2 \]

5. **Substitute the Ratio**:
Now substitute the ratio we found in step 3 into the equation:
\[ \frac{ar(ΔPRQ)}{ar(ΔBCA)} = \left(\frac{3}{1}\right)^2 \]

6. **Calculate the Square**:
Calculate the square of the ratio:
\[ \frac{ar(ΔPRQ)}{ar(ΔBCA)} = \frac{9}{1} = 9 \]

### Final Answer:
\[ \frac{ar(ΔPRQ)}{ar(ΔBCA)} = 9 \]