In triangle ABC, points D and E lie on sides AB and AC respectively such that segment DE is parallel to BC. The ratio of the segments on AB is AD : DB = 2 : 3. What is the ratio of the area of triangle ADE to the area of quadrilateral BDEC?

Introduction:
This problem again uses the similarity of triangles created by drawing a segment parallel to the base of a triangle. The key lies in relating the ratio of corresponding sides to the ratio of areas and then comparing the area of the small top triangle with the remaining quadrilateral at the base.

Given Data / Assumptions:

• Triangle ABC with D on AB and E on AC.
• DE ∥ BC.
• AD : DB = 2 : 3, so AB = AD + DB.
• We must find area(ΔADE) : area(quadrilateral BDEC).

Concept / Approach:
Because DE is parallel to BC, triangle ADE is similar to triangle ABC. The ratio of similarity is AD / AB. Since AB = AD + DB, we get AD : AB and then find the square of this ratio to relate their areas. The quadrilateral BDEC is the portion of triangle ABC left when triangle ADE is removed. We then express the ratio of the small area to the remaining area.

Step-by-Step Solution:
AD : DB = 2 : 3, so AB = AD + DB is in ratio 2 + 3 = 5 parts. Thus AD : AB = 2 : 5, so AD / AB = 2 / 5. Triangles ADE and ABC are similar, so area(ADE) / area(ABC) = (AD / AB)². Therefore area(ADE) / area(ABC) = (2 / 5)² = 4 / 25. Let area(ABC) = A. Then area(ADE) = (4 / 25)A. Area of quadrilateral BDEC = A − (4 / 25)A = (21 / 25)A. Hence area(ADE) : area(BDEC) = (4 / 25)A : (21 / 25)A = 4 : 21.

Verification / Alternative check:
You can check quickly using ratios alone: if total area is considered 25 units, the top triangle is 4 units and the lower quadrilateral is 21 units. The difference 25 − 4 = 21 matches the computation, confirming the ratio 4 : 21.

Why Other Options Are Wrong:
Ratios 4 : 25 or 4 : 29 would imply that the quadrilateral area is larger or smaller in a way that breaks the fixed square relationship between side ratios and area ratios. Only 4 : 21 is consistent with the 2 : 5 side ratio and similar triangles theory.

Common Pitfalls:
A frequent mistake is to use area ratios directly as 2 : 5 instead of squaring the ratio. Another is confusing the ratio AD : DB with AD : AB. Always compute AD : AB first, then square to get the area ratio.

Final Answer:
The required ratio of areas is 4 : 21.