If `triangle`ABC - `triangle`PQR and `(BC)/(QR)` = `1/4` then `(ar(trianglePQR))/(ar(triangleABC))` =

Options

A. `1/4`
B. `4/1`
C. `1/16`
D. `16/1`

Answer

To solve the problem, we need to find the ratio of the areas of two similar triangles, triangle ABC and triangle PQR, given that the ratio of the lengths of the corresponding sides BC and QR is \( \frac{1}{4} \).

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
Since triangle ABC is similar to triangle PQR, the ratio of their corresponding sides is equal. This means that if we know the ratio of one pair of corresponding sides, we can use it to find the ratio of their areas.

**Hint**: Remember that similar triangles have proportional sides.

2. **Given Ratio of Sides**:
We are given that \( \frac{BC}{QR} = \frac{1}{4} \). Here, BC is a side of triangle ABC, and QR is the corresponding side of triangle PQR.

**Hint**: Identify the corresponding sides in similar triangles.

3. **Finding the Inverse Ratio**:
To find the ratio of QR to BC, we take the reciprocal of the given ratio:
\[ \frac{QR}{BC} = \frac{1}{\frac{1}{4}} = 4 \]

**Hint**: When dealing with ratios, the inverse can be useful for finding corresponding relationships.

4. **Using the Area Ratio Theorem**:
The area of similar triangles is proportional to the square of the ratio of their corresponding sides. Therefore, we can express the ratio of the areas as:
\[ \frac{ar(\triangle PQR)}{ar(\triangle ABC)} = \left(\frac{QR}{BC}\right)^2 \]

**Hint**: Recall the theorem that states the area ratio is the square of the side ratio.

5. **Substituting the Ratio**:
Now we substitute the value we found for \( \frac{QR}{BC} \):
\[ \frac{ar(\triangle PQR)}{ar(\triangle ABC)} = \left(4\right)^2 = 16 \]

**Hint**: Make sure to square the ratio when calculating area ratios.

6. **Final Answer**:
Thus, the ratio of the area of triangle PQR to the area of triangle ABC is:
\[ \frac{ar(\triangle PQR)}{ar(\triangle ABC)} = 16 \]

### Conclusion:
The final answer is \( \frac{ar(\triangle PQR)}{ar(\triangle ABC)} = 16 \).