In triangle ABC, points D and E lie on sides AB and AC respectively, and DE is parallel to BC. If AD : DB = 1 : 2 and the area of triangle ABC is 45 square cm, what is the area (in square cm) of quadrilateral BDEC?

Introduction / Context:
This question again uses similar triangles formed when a line segment is drawn parallel to one side of a triangle. It specifically asks for the area of the quadrilateral that lies between the smaller similar triangle at the top and the original base. Understanding how areas scale with side ratios is key here.

Given Data / Assumptions:

• Triangle ABC has D on AB and E on AC.
• DE is parallel to BC.
• AD : DB = 1 : 2, so D divides AB internally in this ratio.
• Area of triangle ABC is 45 square cm.
• We must find the area of quadrilateral BDEC.

Concept / Approach:
Since DE ∥ BC, triangle ADE is similar to triangle ABC. If the ratio of corresponding sides AD : AB is known, then the ratio of their areas is the square of that side ratio. Once we know the area of the smaller triangle ADE, we subtract it from the area of ABC to get the area of quadrilateral BDEC, which is the remaining region between DE and BC.

Step-by-Step Solution:
Step 1: From AD : DB = 1 : 2, we have AB = AD + DB = 1 + 2 = 3 parts. Step 2: Therefore, AD / AB = 1 / 3. Step 3: Because DE ∥ BC, triangles ADE and ABC are similar. Step 4: The ratio of their areas equals the square of the ratio of corresponding sides: area(ADE) / area(ABC) = (AD / AB)^2 = (1 / 3)^2 = 1 / 9. Step 5: Given area(ABC) = 45 square cm, we have area(ADE) = (1 / 9) * 45 = 5 square cm. Step 6: The area of quadrilateral BDEC is the remaining area after removing triangle ADE from the whole triangle ABC. Step 7: So area(BDEC) = area(ABC) − area(ADE) = 45 − 5 = 40 square cm.

Verification / Alternative check:
We can also reason with fractions of area. Triangle ADE occupies 1 / 9 of the area of ABC, so the remaining portion, which is quadrilateral BDEC, must occupy 8 / 9 of the area. Then 8 / 9 of 45 is (8 * 45) / 9 = 8 * 5 = 40. This confirms the computed value of 40 square cm and shows the consistency of the approach.

Why Other Options Are Wrong:
20 and 15 square cm are too small; they would correspond to a smaller fraction of the original triangle than the 8 / 9 that the quadrilateral actually represents. 30 square cm would correspond to 2 / 3 of the total area, which does not match the correct similarity ratio. 25 square cm does not arise from any simple fraction of 45 given the 1 : 2 division of AB, so it cannot be correct.

Common Pitfalls:
Sometimes students mistakenly use the ratio 1 : 2 directly for area instead of converting it to AD : AB and then squaring. Others forget that AB is the sum of AD and DB, leading to a wrong side ratio. Another error is to assume that the quadrilateral has the same area as triangle ADE or as triangle ABC, which is clearly not supported by the similarity relations. Always carefully compute the side ratio with respect to the entire side AB and then square it to get the area ratio.

Final Answer:
The area of quadrilateral BDEC is 40 square cm.