In triangle ABC, points D and E lie on sides AB and AC respectively, with DE parallel to BC. If AD : DB = 2 : 5 and the area of triangle ADE is 8 square cm, what is the ratio of the area of triangle ADE to the area of quadrilateral BDEC?

Introduction / Context:
This question checks your understanding of similar triangles and how their areas scale with side lengths. When a line segment is drawn parallel to one side of a triangle, it creates a smaller, similar triangle, and area ratios can be derived directly from the ratio of corresponding sides. This is a common pattern in geometry and aptitude exams.

Given Data / Assumptions:

• Triangle ABC has D on AB and E on AC.
• DE is parallel to BC.
• AD : DB = 2 : 5.
• The area of triangle ADE is 8 square cm.
• We need the ratio of area(ADE) : area(BDEC), where BDEC is the quadrilateral formed between DE and BC.

Concept / Approach:
When a line through a vertex is drawn parallel to the opposite side of a triangle, the smaller triangle is similar to the original triangle. If the ratio of a pair of corresponding sides is k, the ratio of their areas is k^2. Here, AD corresponds to AB in triangles ADE and ABC. Once we know the ratio of areas, we can find the area of the whole triangle ABC and then subtract the area of ADE to get the area of quadrilateral BDEC.

Step-by-Step Solution:
Step 1: From AD : DB = 2 : 5, we have AB = AD + DB = 2 + 5 = 7 parts. Step 2: So AD / AB = 2 / 7. Step 3: Since DE ∥ BC, triangles ADE and ABC are similar. Step 4: The ratio of their areas is the square of the ratio of corresponding sides: area(ADE) / area(ABC) = (AD / AB)^2 = (2 / 7)^2 = 4 / 49. Step 5: Let area(ABC) = X. Then 8 / X = 4 / 49, so X = 8 * 49 / 4 = 8 * 12.25 = 98 square cm. Step 6: The area of quadrilateral BDEC is area(ABC) − area(ADE) = 98 − 8 = 90 square cm. Step 7: So the desired ratio area(ADE) : area(BDEC) is 8 : 90, which simplifies by dividing both numbers by 2 to 4 : 45.

Verification / Alternative check:
We can verify the ratio directly from areas. area(ADE) / area(ABC) = 4 / 49 and area(BDEC) / area(ABC) = 45 / 49, because the remaining part of the triangle occupies the rest of the area. Then area(ADE) : area(BDEC) = 4 / 49 : 45 / 49 = 4 : 45, which agrees with our computation based on actual numbers. This confirms the correctness of the ratio.

Why Other Options Are Wrong:
45 : 4 and 45 : 8 invert or distort the correct ratio, implying the quadrilateral is smaller than triangle ADE, which is impossible since it occupies most of the original triangle. 8 : 45 is close numerically to the simplified ratio but does not reduce to an integer ratio with area(ABC) consistent at 98 square cm. 1 : 4 would suggest that ADE is much smaller in area relative to BDEC than is actually the case in this configuration.

Common Pitfalls:
A common mistake is to use the given side ratio 2 : 5 directly for area ratios instead of squaring the scale factor 2 / 7. Another frequent error is forgetting that AB is the sum of AD and DB, leading to an incorrect scale factor. Always compute the full side ratio between corresponding sides first, then square it for area ratios when triangles are similar.

Final Answer:
The required ratio of areas is 4 : 45.