In triangle ΔABC, points D and E lie on sides AB and AC respectively, and DE is parallel to BC. If AD : DB = 2 : 5 and the area of ΔADE is 8 cm², then what is the area (in square centimetres) of quadrilateral BDEC?

Introduction / Context:
This geometry question uses similarity of triangles and area ratios. When a line segment joins two sides of a triangle parallel to the third side, the smaller triangle formed is similar to the original triangle. This allows us to relate areas through the square of a linear scale factor, a very important concept in geometry and mensuration problems.

Given Data / Assumptions:

• Triangle ΔABC with D on AB and E on AC.
• DE ∥ BC.
• AD : DB = 2 : 5, so AB is divided in ratio 2 : 5.
• Area of ΔADE is 8 cm².
• We need the area of quadrilateral BDEC.

Concept / Approach:
Because DE is parallel to BC, triangles ADE and ABC are similar. The ratio of similarity between corresponding sides is AD / AB. Since AB = AD + DB, we can compute this ratio. The area scale factor between the smaller and larger triangle is the square of the linear scale factor. Once we find the total area of ΔABC, subtracting the area of ΔADE gives the area of quadrilateral BDEC, which is the region between DE and BC.

Step-by-Step Solution:
Given AD : DB = 2 : 5, so let AD = 2k and DB = 5k. Then AB = AD + DB = 2k + 5k = 7k. The linear scale factor from ΔADE to ΔABC is AD / AB = 2k / 7k = 2/7. Areas of similar triangles scale by the square of the linear factor. Therefore, Area(ΔADE) / Area(ΔABC) = (2/7)^2 = 4/49. Given Area(ΔADE) = 8 cm², so 8 / Area(ΔABC) = 4/49. Solving gives Area(ΔABC) = 8 * 49 / 4 = 8 * 12.25 = 98 cm². Quadrilateral BDEC is the region ΔABC minus ΔADE, so Area(BDEC) = 98 − 8 = 90 cm².

Verification / Alternative check:
We can view ABC as composed of the small triangle ADE plus trapezium BDEC. The ratio of corresponding heights from A to BC and from A to DE is also 7 : 2, consistent with the side ratio. Hence, the large area should be larger than the small one by a factor of 49/4 = 12.25. Multiplying 8 by 12.25 gives 98, which is consistent. Subtracting the known 8 cm² area of ΔADE from the total again gives 90 cm², confirming the calculation.

Why Other Options Are Wrong:
If one mistakenly uses 2/5 as the scale factor, incorrect areas such as 80 cm² or 86 cm² can appear. Using a linear ratio directly without squaring can also produce options like 94 cm² or 98 cm² for the quadrilateral instead of the whole triangle. Only 90 cm² correctly represents the remaining area after removing ΔADE from ΔABC.

Common Pitfalls:
A frequent error is to forget that area scales with the square of the similarity ratio, not linearly. Another mistake is to mix up the structure and subtract in the wrong order, or to interpret 2 : 5 as the ratio between DE and BC directly. Carefully identifying corresponding sides and remembering that area ratio equals the square of side ratio are essential for accuracy.

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