In triangle ABC, which is equilateral, points D and E are the midpoints of sides AB and BC respectively. What is the ratio of the area of triangle ABC to the area of the quadrilateral ADEC (called a trapezium in the statement)?

Introduction / Context:
This is a geometry question about area ratios inside an equilateral triangle. When midpoints of sides are joined or used to form sub figures, the resulting smaller triangles and quadrilaterals often have areas in simple ratios relative to the original triangle. Recognising such patterns saves time in exams, especially in questions on equilateral triangles and mid segments.

Given Data / Assumptions:
- Triangle ABC is equilateral, so all sides and all angles are equal.
- D is the midpoint of side AB.
- E is the midpoint of side BC.
- Quadrilateral ADEC is formed by points A, D, E and C inside the triangle.
- We need the ratio area of triangle ABC : area of quadrilateral ADEC.

Concept / Approach:
An equilateral triangle has many symmetry properties. When midpoints of sides are used, line segments joining them create smaller congruent triangles. A helpful approach is to assign a side length to the equilateral triangle, use coordinate geometry or known area formulas to compute exact areas and then take the ratio. Another approach is to use similar triangles and proportionality, because segments joining midpoints often create triangles that are similar to the original triangle with predictable area ratios.

Step-by-Step Solution:
Step 1: Assume the side length of equilateral triangle ABC is s units for convenience. Step 2: The area of an equilateral triangle with side s is (√3 / 4) * s^2, so area of ABC is (√3 / 4) * s^2. Step 3: Place the triangle in a coordinate system. Let A be at (0, 0), B at (s, 0) and C at (s / 2, (√3 / 2) * s). Step 4: D, the midpoint of AB, has coordinates ((0 + s) / 2, 0) = (s / 2, 0). E, the midpoint of BC, has coordinates ((s + s / 2) / 2, (0 + (√3 / 2) * s) / 2) = (3s / 4, (√3 / 4) * s). Step 5: Use a standard polygon area method or known results to find that the area of quadrilateral ADEC is (3 / 4) of the area of triangle ABC. Hence area ABC : area ADEC = 4 : 3.

Verification / Alternative check:
An alternative more conceptual check is to observe that joining the midpoints in an equilateral triangle partitions it into four small congruent equilateral triangles, each of area (1 / 4) of the original. The quadrilateral ADEC covers three of these smaller parts when you visualise the figure, leaving one small triangle outside near vertex B. Thus the quadrilateral ADEC has an area equal to 3 / 4 of the original triangle, again giving the ratio 4 : 3.

Why Other Options Are Wrong:
5 : 3, 4 : 1, 8 : 5 and 3 : 2 do not match the geometric partitioning implied by midpoints in an equilateral triangle. None of these ratios correspond to a pattern where one small triangular region is exactly one quarter of the whole, with the remaining region being three quarters.

Common Pitfalls:
A common mistake is to assume that each new segment always divides the triangle into halves or equal parts, which is not true in general. Another pitfall is misidentifying which small triangles are included inside the quadrilateral and which one remains outside. Drawing a clear diagram and, if needed, marking equal side segments can prevent such confusion.

Final Answer:
The required ratio of areas is 4 : 3.