In triangle ABC, points D and E lie on sides AB and AC respectively. Line segment DE is drawn parallel to BC. If the ratio AD : DB is 2 : 3, then what is the ratio of the area of triangle ADE to the area of quadrilateral BDEC?

Introduction / Context:
This geometry problem tests the concept of similarity of triangles and how areas scale when corresponding sides are in a fixed ratio. When a line is drawn parallel to one side of a triangle and intersects the other two sides, a smaller similar triangle is formed inside the larger one. The question asks for the ratio of the area of this smaller triangle to the area of the remaining quadrilateral inside the original triangle.

Given Data / Assumptions:

• Triangle ABC is a general triangle.
• Point D lies on side AB and point E lies on side AC.
• Segment DE is parallel to side BC.
• The ratio AD : DB is 2 : 3.
• We are asked to find area(ΔADE) : area(quadrilateral BDEC).

Concept / Approach:
When a line segment is drawn parallel to the base of a triangle and cuts the other two sides, the smaller triangle formed at the top is similar to the original triangle. If the ratio of corresponding sides of the smaller to the larger triangle is k, then the ratio of their areas is k^2. The area of the quadrilateral BDEC is simply the area of the big triangle minus the area of the small triangle.

Step-by-Step Solution:
Step 1: Given AD : DB = 2 : 3, the total length AB is divided into 2 + 3 = 5 equal parts.Step 2: The fraction of AB that corresponds to AD is AD / AB = 2 / 5.Step 3: Because DE is parallel to BC, triangle ADE is similar to triangle ABC. The similarity ratio of corresponding sides is AD / AB = 2 / 5, which is also AE / AC.Step 4: Therefore, area(ΔADE) : area(ΔABC) = (2 / 5)^2 = 4 / 25.Step 5: Let the area of ΔABC be A. Then area(ΔADE) = (4 / 25)A.Step 6: The area of quadrilateral BDEC is the remaining part: A - (4 / 25)A = (21 / 25)A.Step 7: Hence, area(ΔADE) : area(quadrilateral BDEC) = (4 / 25)A : (21 / 25)A = 4 : 21.

Verification / Alternative check:
You can verify the logic by assuming a convenient numeric value for AB, such as AB = 5 units, so that AD = 2 units and DB = 3 units. Then, use similarity to find relative heights, compute approximate areas for the two triangles, and check that the ratio still comes out as 4 : 21. Any consistent assignment will lead to the same ratio, confirming correctness.

Why Other Options Are Wrong:
4 : 25 is the ratio of the small triangle to the whole triangle, not to the quadrilateral. 4 : 29 and 5 : 21 do not arise from squaring the side ratio and subtracting from the total. 4 : 9 would correspond to a different side ratio that is not consistent with AD : DB = 2 : 3.

Common Pitfalls:
A common mistake is to take the side ratio 2 : 5 directly as the area ratio, forgetting to square the factor when moving from sides to areas. Another frequent error is to confuse the ratio of the small triangle to the big triangle with the ratio of the small triangle to the remaining region. Careful stepwise use of similarity and area subtraction avoids these issues.

Final Answer:
The required ratio of the area of triangle ADE to the area of quadrilateral BDEC is 4 : 21.