In triangle △ABC, points D and E are the midpoints of AB and AC respectively. What is the ratio of areas ar(△ADE) : ar(△ABC)?

Introduction / Context:This is a standard midpoint/similarity fact in triangles. Joining a vertex to the midpoints of the two adjacent sides creates a triangle similar to the original with a linear scale factor 1/2, so its area is 1/4 of the original.

Given Data / Assumptions:

• D is the midpoint of AB and E is the midpoint of AC.
• Segment DE is parallel to BC (Midpoint Theorem).
• △ADE is formed by joining A to D and E.

Concept / Approach:

• By the Midpoint Theorem, DE ∥ BC and AD/AB = AE/AC = 1/2.
• Thus △ADE ~ △ABC with similarity ratio (linear) k = 1/2.
• Area scales as k^2, hence area(△ADE) = (1/2)^2 * area(△ABC) = 1/4 area(△ABC).

Step-by-Step Solution:

Verification / Alternative check:Pick coordinates: A(0,0), B(2,0), C(0,2). Then D(1,0), E(0,1). Area(△ABC)=2, Area(△ADE)=0.5 ⇒ ratio 1:4.

Why Other Options Are Wrong:

• 1:2 and 3:4 overestimate the smaller triangle.
• 1:8 underestimates; that would correspond to a linear scale of 1/√8, not 1/2.
• None of these: Incorrect because 1:4 is exact.

Common Pitfalls:

• Confusing length ratios (1/2) with area ratios (1/4).
• Forgetting that DE ∥ BC guarantees similarity.

Final Answer:1 : 4