PBA and PDC are two secants. AD is the diameter of the circle with centre at O. \( \angle A=40° \), \( \angle P=20° \). Find the measure of \( \angle DBC \).
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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.
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) \( \frac{8}{17} \).
