Midpoint (medial) triangle area fraction: In triangle ABC, points D, E, F are midpoints of BC, CA, and AB, respectively. If area(ΔABC) = 36 sq m, find area(ΔDEF).

Introduction / Context:The triangle formed by joining the midpoints of the sides of a triangle is called the medial triangle. It is similar to the original triangle, and its area is a fixed fraction of the original.

Given Data / Assumptions:

• D, E, F are midpoints of BC, CA, AB
• area(ΔABC) = 36 sq m
• Standard midpoint (mid-segment) theorem applies

Concept / Approach:The medial triangle is similar to the original triangle with linear scale factor 1/2 (each side is half). Area scales as the square of the linear factor → (1/2)^2 = 1/4.

Step-by-Step Solution:Scale factor (length) = 1/2Area factor = (1/2)^2 = 1/4area(ΔDEF) = (1/4) * area(ΔABC) = (1/4) * 36 = 9 sq m

Verification / Alternative check:Coordinate geometry check: place ABC conveniently and compute midpoints; the determinant-based area confirms the 1/4 rule.

Why Other Options Are Wrong:12 sq m equals 1/3, not the correct 1/4; 18 sq m is half; 19 sq m has no basis in the midpoint theorem.

Common Pitfalls:Confusing side-halving with area-halving; area reduces by the square of the scale factor.

Final Answer:9 sq m