In triangle ABC, points D and E lie on sides AB and AC respectively, and segment DE is drawn parallel to side BC. The segments on side AB satisfy the ratio AD : DB = 2 : 5. The area of the smaller triangle ADE is 8 cm². Using properties of similar triangles and area ratios, what is the area (in cm²) of the quadrilateral BDEC (that is, the region between DE and BC)?

Introduction:
This geometry question tests the concept of similar triangles formed by a line drawn parallel to the base of a triangle, and how their areas are related to the ratios of corresponding sides. Once similarity is understood, the area of the quadrilateral can be found by subtracting the smaller triangle area from the larger triangle area.

Given Data / Assumptions:

• In triangle ABC, D lies on AB and E lies on AC.
• DE is parallel to BC.
• AD : DB = 2 : 5, so AB is divided internally in this ratio.
• Area of triangle ADE = 8 cm².
• We must find the area of quadrilateral BDEC.

Concept / Approach:
When a line segment is drawn parallel to one side of a triangle and cutting the other two sides, the smaller triangle formed at the vertex is similar to the original triangle. The ratio of similarity is the ratio of corresponding sides, and the ratio of their areas is the square of the ratio of their corresponding sides. After obtaining the area of the whole triangle ABC, the quadrilateral area is simply the difference between the total area and the area of the smaller triangle.

Step-by-Step Solution:
Let AB = AD + DB. Given AD : DB = 2 : 5, so AD : AB = 2 : (2 + 5) = 2 : 7. Thus, the ratio of similarity (ADE to ABC) is AD / AB = 2 / 7. Area scale factor = (2 / 7)² = 4 / 49. Let area of triangle ABC = K. Then area(ADE) = (4 / 49) * K. Given area(ADE) = 8, so 8 = (4 / 49) * K. Solve for K: K = 8 * (49 / 4) = 2 * 49 = 98 cm². Quadrilateral BDEC is the remaining part: area(BDEC) = area(ABC) − area(ADE) = 98 − 8 = 90 cm².

Verification / Alternative check:
As a ratio, area(ADE) : area(ABC) = 4 : 49. That means area(BDEC) : area(ADE) = (49 − 4) : 4 = 45 : 4. If area(ADE) is 8, then area(BDEC) = (45 / 4) * 8 = 45 * 2 = 90 cm², which confirms our previous result.

Why Other Options Are Wrong:
Values 86, 94, and 98 cm² do not satisfy the required area ratio. In particular, 98 cm² is the total area of triangle ABC, not the quadrilateral. Any value other than 90 would break the fixed similarity ratio implied by AD : AB.

Common Pitfalls:
Learners sometimes assume that area ratios are directly proportional to side ratios instead of the square of side ratios. Another error is to forget that AB is the sum AD + DB when forming the 2 : 7 ratio. Being precise with ratios and squaring for area is essential.

Final Answer:
The area of quadrilateral BDEC is 90 cm².