Question

In triangle ABC, two points P and Q are on AB and BC respectively such that \( \frac{AP}{BP} = 1 : 4 \) and \( \frac{BQ}{CQ} = 2 : 3 \). The ratio of areas of triangle BPQ and the quadrilateral PQCA is:

• A. \( \frac{8}{25} \)
• B. \( \frac{17}{25} \)
• C. \( \frac{9}{17} \)
• D. \( \frac{8}{17} \)

Hint:

When dealing with problems involving ratios of areas in triangles, remember that the areas are proportional to the square of the corresponding sides. In cases where points divide sides of a triangle, use these ratios to find the area relationships.

Answer 1

The Correct Option is D

Step 1: Understand the given information.
- The points \( P \) and \( Q \) divide the sides \( AB \) and \( BC \) in the ratios \( \frac{AP}{BP} = 1 : 4 \) and \( \frac{BQ}{CQ} = 2 : 3 \), respectively.
- The total area of triangle ABC can be broken into smaller areas based on these ratios. Step 2: Apply the area of triangles and proportionality.
- The area of triangle \( BPQ \) is proportional to the areas of triangles formed by the division of the sides.
- Using the formula for the areas of triangles with proportional sides, we can find the ratio of areas of triangle BPQ and the quadrilateral PQCA. Step 3: Calculate the ratio of areas.
- Based on the properties of triangles and the given ratios, the area of triangle BPQ and quadrilateral PQCA are related by the ratio \( \frac{8}{17} \).
Final Answer: The correct answer is (d) \( \frac{8}{17} \).