In triangle ABC, points D and E lie on sides AB and AC respectively such that DE is parallel to BC. If AD:DB = 2:3 and the area of triangle ABC is 100 square centimetres, what is the area of quadrilateral BDEC in square centimetres?

Introduction / Context:
This geometry question uses similar triangles formed by a line parallel to the base of a triangle. When DE is parallel to BC, triangle ADE is similar to triangle ABC. Knowing the ratio of corresponding sides allows us to find the ratio of areas. Once the area of the smaller triangle ADE is known, subtracting it from the area of ABC gives the area of the remaining quadrilateral BDEC.

Given Data / Assumptions:

• Triangle ABC has area 100 sq cm.
• D lies on AB, E lies on AC.
• DE is parallel to BC.
• AD:DB = 2:3, so AB is divided in the ratio 2:3.
• Triangles ADE and ABC are similar.

Concept / Approach:
If two triangles are similar, the ratio of their areas equals the square of the ratio of corresponding sides. Here AD corresponds to AB. Since AD:DB = 2:3, AB = AD + DB = 2 + 3 = 5 equal parts, so AD:AB = 2:5. Therefore, area(ADE) : area(ABC) = (2/5)^2 = 4/25. With area(ABC) known, area(ADE) can be computed, and area(BDEC) = area(ABC) - area(ADE).

Step-by-Step Solution:
AD:DB = 2:3, so AB consists of 2 + 3 = 5 equal parts.Therefore AD:AB = 2:5.Triangles ADE and ABC are similar since DE is parallel to BC.Area ratio = (corresponding side ratio)^2 = (AD / AB)^2 = (2/5)^2 = 4/25.Area of triangle ABC = 100 sq cm.Area of triangle ADE = (4/25) * 100 = 400/25 = 16 sq cm.Area of quadrilateral BDEC = area(ABC) - area(ADE) = 100 - 16 = 84 sq cm.

Verification / Alternative check:
As a check, the remaining area (84 sq cm) is significantly larger than the small triangle area (16 sq cm), which makes sense because ADE sits near the vertex A and is smaller than ABC. The fraction 4/25 of the whole is 16%, leaving 84% of the area as BDEC, which matches 84 out of 100.

Why Other Options Are Wrong:
16 sq cm is the area of triangle ADE, not the quadrilateral. 25 sq cm arises if someone mistakenly uses 1/4 instead of 4/25 as the area ratio. 75 sq cm results from subtracting 25 instead of 16. 64 sq cm comes from using (2/5) for area ratio without squaring it, giving area(ADE) = 40 and remaining area 60, which is inconsistent with the correct similarity rules.

Common Pitfalls:
Students often forget that area ratios are squares of side ratios, not the same ratios. Others misinterpret AD:DB as AD:AB. Some may try to use height ratios directly without establishing correct similarity. Keeping track of the full side AB as 5 parts and understanding that AD is 2 of those parts is critical for solving the problem correctly.

Final Answer:
84 sq cm