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:
AD = AB/2, AE = AC/2 ⇒ similarity ratio = 1/2Area ratio = (1/2)^2 = 1/4
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
Discussion & Comments