It is given that `Delta ABC - Delta PQR`, with `(BC)/(QR) = (1)/(4)`. Then `(ar Delta PQR)/(ar Delta ABC)` is equal to:

Options

A. 16
B. 3
C. `(1)/(3)`
D. `(1)/(9)`

Answer

To solve the problem, we need to find the ratio of the areas of triangles PQR and ABC given that the triangles are similar and the ratio of the lengths of one pair of corresponding sides is given.

### Step-by-Step Solution:

1. **Understand the Given Information:**
We know that triangles ABC and PQR are similar. The ratio of the lengths of the sides BC and QR is given as:
\[ \frac{BC}{QR} = \frac{1}{4} \]

2. **Use the Property of Similar Triangles:**
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore:
\[ \frac{ar \Delta PQR}{ar \Delta ABC} = \left(\frac{QR}{BC}\right)^2 \]

3. **Substitute the Given Ratio:**
We know that:
\[ \frac{BC}{QR} = \frac{1}{4} \]
This implies:
\[ \frac{QR}{BC} = 4 \quad \text{(by taking the reciprocal)} \]

4. **Calculate the Area Ratio:**
Now, substituting this into the area ratio formula:
\[ \frac{ar \Delta PQR}{ar \Delta ABC} = \left(4\right)^2 = 16 \]

5. **Final Answer:**
Thus, the ratio of the areas of triangle PQR to triangle ABC is:
\[ \frac{ar \Delta PQR}{ar \Delta ABC} = 16 \]