In triangle ABC, points D, E and F are the midpoints of sides BC, CA and AB respectively. What is the ratio of the area of triangle DEF to the area of triangle ABC?

Introduction / Context:
This geometry question is about the medial triangle of a given triangle. When you join the midpoints of the sides of a triangle, you get a smaller triangle inside it. The problem asks you to recall and apply the relationship between the areas of the original triangle and this medial triangle. Understanding this concept is helpful in many area and similarity problems.

Given Data / Assumptions:

• Triangle ABC is given.
• D is the midpoint of side BC.
• E is the midpoint of side CA.
• F is the midpoint of side AB.
• Triangle DEF is formed by joining these midpoints.
• We must find the ratio area(DEF) : area(ABC).

Concept / Approach:
When the midpoints of the sides of a triangle are joined, the resulting triangle DEF is called the medial triangle. It is similar to the original triangle ABC and each side of DEF is half the length of the corresponding side of ABC. Since the ratio of similarity of corresponding sides is 1 : 2, the ratio of areas is the square of this ratio, that is (1 / 2)^2 = 1 / 4. Therefore the area of the medial triangle is one quarter of the area of the original triangle.

Step-by-Step Solution:
Step 1: Observe that D, E and F are midpoints of BC, CA and AB respectively. Step 2: When you join these midpoints, triangle DEF is formed inside triangle ABC. Step 3: Each side of triangle DEF is parallel to a side of triangle ABC and is exactly half its length. Step 4: Therefore triangles DEF and ABC are similar with side ratio 1 : 2. Step 5: The ratio of areas of similar triangles is the square of the ratio of corresponding sides. Step 6: So area(DEF) : area(ABC) = (1 / 2)^2 : 1 = 1 / 4 : 1. Step 7: Hence the ratio area(DEF) : area(ABC) is 1 / 4.

Verification / Alternative check:
To verify, you can take a simple coordinate example. Let triangle ABC have vertices A(0, 0), B(2, 0) and C(0, 2). Its area is 2 square units. The midpoints are F(1, 0), D(1, 1) and E(0, 1). Triangle DEF has vertices (1, 1), (0, 1) and (1, 0). Its area is 0.5 square units, which is exactly one quarter of 2. This numerical example confirms the general area ratio of 1 : 4.

Why Other Options Are Wrong:
1 / 2: This would correspond to a side ratio of 1 : √2, which is not true here.
1 / 8 and 1 / 16: These imply much smaller triangles relative to ABC than the actual medial triangle, and do not match the side length relationship of 1 : 2.
3 / 4: This would suggest that the medial triangle occupies most of the area of ABC, which is not correct.

Common Pitfalls:
Students may mistakenly think each side being half implies each area is also half. However, area scales with the square of the side length, not directly with it. Another common error is failing to recognise that DEF is always similar to ABC. Remember that joining midpoints in a triangle always creates a medial triangle with area exactly one quarter of the original.

Final Answer:
The ratio of the areas is area(DEF) : area(ABC) = 1 / 4.