In triangle ABC, points D and E lie on sides AB and AC respectively, with DE parallel to BC. If AD:DB = 1:4 and the area of triangle ADE is 6 sq cm, what is the ratio of area of triangle ADE to the area of quadrilateral BDEC?

Introduction / Context:
This problem again uses similar triangles formed by a line drawn parallel to the base of a triangle. When such a line is drawn, it produces a smaller, similar triangle and a trapezoidal region (here the quadrilateral BDEC). Understanding how the areas of the similar triangles scale with the square of the side ratio allows us to compute area ratios between the small triangle and the remaining quadrilateral.

Given Data / Assumptions:

• D lies on AB and E lies on AC of triangle ABC.
• DE is parallel to BC, so triangle ADE is similar to triangle ABC.
• The ratio AD : DB is 1 : 4.
• The area of triangle ADE is 6 square centimetres.
• We must find the ratio area(ADE) : area(BDEC).

Concept / Approach:
Because DE is parallel to BC, triangles ADE and ABC are similar. If AD : AB is the linear ratio, then the area ratio of the smaller to the larger triangle is the square of that ratio. Once we know the areas of the two triangles, we can find the area of the quadrilateral by subtraction, and then form the required ratio of the small triangle to the quadrilateral.

Step-by-Step Solution:
Step 1: From AD : DB = 1 : 4, we find AB = AD + DB = 1 + 4 = 5 parts. Step 2: Therefore AD / AB = 1 / 5. Step 3: The area ratio of triangle ADE to triangle ABC is (AD / AB)^2 = (1 / 5)^2 = 1 / 25. Step 4: Let the area of triangle ABC be A. Then area(ADE) = (1 / 25) * A. Step 5: We are given area(ADE) = 6 sq cm, so (1 / 25) * A = 6. Step 6: Solve for A: A = 6 * 25 = 150 sq cm. Step 7: The area of quadrilateral BDEC is area(ABC) - area(ADE) = 150 - 6 = 144 sq cm. Step 8: Now form the ratio area(ADE) : area(BDEC) = 6 : 144. Step 9: Simplify 6 : 144 by dividing both numbers by 6 to get 1 : 24.

Verification / Alternative check:
We can confirm the area ratio between the two triangles is 1 : 25. That means the trapezoidal region (quadrilateral BDEC) should have 24 times the area of the small triangle, since 1 part belongs to triangle ADE and 24 parts to the remaining region. Since area(ADE) = 6, the quadrilateral should have area 6 * 24 = 144, matching the computed value. This confirms that the ratio 1 : 24 is correct.

Why Other Options Are Wrong:

• 1:12, 1:6, 1:16, and 1:20: These ratios do not match the squared side scaling. They would imply different relationships between AD and AB or an incorrect area of the entire triangle, contradicting the similarity conditions and given area.

Common Pitfalls:
Some students mistakenly take the area ratio as 1 : 5 instead of 1 : 25, forgetting that area scales with the square of the side ratio. Others mix up which region is the quadrilateral and which is the whole triangle. Clearly separating the full triangle area from the small similar triangle area and then subtracting is the key to obtaining the correct ratio.

Final Answer:
Therefore, the required ratio of the area of triangle ADE to the area of quadrilateral BDEC is 1:24.