In triangle ΔABC, points D and E lie on sides AB and AC respectively. Segment DE is parallel to BC. If AD : DB = 2 : 5 and the area of ΔABC is 98 sq cm, what is the area (in sq cm) of quadrilateral BDEC?

Introduction / Context:
This geometry question involves similar triangles formed by drawing a line parallel to the base of a triangle. You are given a ratio on one side and the total area of the main triangle, and you must find the area of a quadrilateral formed between the parallel line and the base. It tests your mastery of area ratios in similar triangles and simple subtraction of areas.

Given Data / Assumptions:

• ΔABC is a triangle.
• D lies on AB and E lies on AC.
• DE is parallel to BC.
• AD : DB = 2 : 5.
• Area of ΔABC = 98 sq cm.
• We need the area of quadrilateral BDEC.

Concept / Approach:
When a line segment is drawn parallel to the base of a triangle and cuts the other two sides, the smaller triangle at the top is similar to the original triangle. The ratio of similarity equals the ratio of corresponding sides. Because area scales as the square of the linear scale factor, we can find the area of the smaller top triangle and subtract it from the total area to get the area of the remaining quadrilateral BDEC.

Step-by-Step Solution:
Given AD : DB = 2 : 5. Then AB = AD + DB corresponds to 2 + 5 = 7 parts, so AD : AB = 2 : 7. Since DE is parallel to BC, triangle ADE is similar to triangle ABC. The linear scale factor (ADE : ABC) = AD / AB = 2 / 7. Area scale factor for similar triangles is the square of the side ratio, so Area(ADE) / Area(ABC) = (2 / 7)^2 = 4 / 49. Area of ΔADE = (4 / 49) * 98 = 4 * 2 = 8 sq cm. Area of quadrilateral BDEC = Area(ABC) - Area(ADE) = 98 - 8 = 90 sq cm.

Verification / Alternative check:
It is useful to cross check the fractions. If the top triangle occupies 4/49 of the full area, then the remaining lower region, which includes quadrilateral BDEC, should occupy 45/49 of the full area. 45/49 of 98 = 45 * 2 = 90 sq cm, which matches the earlier computation exactly, confirming the answer.

Why Other Options Are Wrong:
98: This is the area of the entire triangle, not the quadrilateral. 94 and 86: These correspond to subtracting incorrect fractions of the total area, indicating a wrong area ratio. 88: Another incorrect subtraction, not supported by the similarity and area ratio rules.

Common Pitfalls:
Using AD : DB directly as the area ratio instead of converting to AD : AB. Forgetting to square the side ratio when moving to area ratios. Subtracting the wrong part of the area or confusing triangle ADE with the quadrilateral.

Final Answer:
The area of quadrilateral BDEC is 90 sq cm