In triangle ABC, points D and E lie on sides AB and AC respectively, with DE parallel to BC. If AD:DB = 2:3 and the area of triangle ADE is 4 sq cm, what is the area of quadrilateral BDEC (in sq cm)?

Introduction / Context:
This geometry problem explores the relationship between similar triangles formed by drawing a line parallel to the base of a triangle. When a line segment joins two sides of a triangle and is parallel to the third side, it creates a smaller triangle similar to the original one. Their areas are related by the square of the ratio of corresponding sides. This idea is very useful in questions involving areas of triangles and quadrilaterals inside a larger triangle.

Given Data / Assumptions:

• D is on side AB and E is on side AC of triangle ABC.
• DE is parallel to BC, so triangle ADE is similar to triangle ABC.
• The ratio AD : DB is 2 : 3.
• The area of triangle ADE is 4 square centimetres.
• We need the area of quadrilateral BDEC.

Concept / Approach:
Since DE is parallel to BC, triangle ADE is similar to triangle ABC. The ratio of similarity is based on corresponding sides. If AD : AB = k, then the ratio of the areas is k^2, because area scales with the square of the linear scale factor. Once we find the ratio of the areas of the small triangle to the large triangle, we can find the area of the full triangle ABC and then subtract the area of triangle ADE to obtain the area of quadrilateral BDEC.

Step-by-Step Solution:
Step 1: From AD : DB = 2 : 3, the whole side AB is AD + DB = 2 + 3 = 5 parts. Step 2: Therefore AD / AB = 2 / 5. Step 3: Because triangles ADE and ABC are similar, the ratio of their areas is (AD / AB)^2 = (2 / 5)^2. Step 4: Compute the area ratio: (2 / 5)^2 = 4 / 25. Step 5: Let the area of triangle ABC be A. Then 4 / 25 of A equals the area of triangle ADE, which is 4 sq cm. Step 6: So (4 / 25) * A = 4. Therefore A = 4 * 25 / 4 = 25 sq cm. Step 7: Quadrilateral BDEC occupies the remaining area: area(BDEC) = area(ABC) - area(ADE) = 25 - 4 = 21 sq cm.

Verification / Alternative check:
We can also think in terms of a scale factor. The side of the larger triangle compared to the smaller is AB / AD = 5 / 2. The area of the larger triangle is then (5 / 2)^2 = 25 / 4 times the area of the smaller. With area of ADE equal to 4, area of ABC equals 25 sq cm again. Subtracting confirms the quadrilateral has area 21 sq cm, consistent with the previous method.

Why Other Options Are Wrong:

• 25: This is the area of the entire triangle ABC, not of the quadrilateral.
• 5 and 9: These are too small and ignore the correct area scaling relationship between similar triangles.
• 16: This might come from incorrect subtraction or an incorrect area ratio but does not satisfy the similarity conditions.

Common Pitfalls:
A common error is to assume that the area scales in direct proportion to the side ratio instead of using the square of the ratio. Another mistake is misreading the ratio AD : DB and using AD : AB incorrectly. Carefully combining the ratio of sides, square relation for areas, and subtraction from the full triangle helps avoid these issues.

Final Answer:
Therefore, the area of quadrilateral BDEC is 21 sq cm.