Parallel to base halves area – find AX/AB: In ΔABC, a line XY ∥ AC divides the triangle into two equal-area parts. What is AX as a fraction of AB?

Introduction / Context:
In a triangle, a line drawn parallel to a side creates a smaller similar triangle at the vertex. Areas of similar triangles scale as the square of the similarity ratio.

Given Data / Assumptions:

• XY ∥ AC in ΔABC.
• Area(ΔAXY) = (1/2)·Area(ΔABC).
• AX/AB is the similarity (linear) ratio between ΔAXY and ΔABC.

Concept / Approach:
For similar triangles, (Area ratio) = (Linear ratio)^2. If the area ratio is 1/2, then the linear ratio is √(1/2) = 1/√2.

Step-by-Step Solution:
Let k = AX/AB.Given Area(AXY)/Area(ABC) = 1/2 = k^2 ⇒ k = 1/√2.

Verification / Alternative check:
With k = 1/√2, the remaining trapezoid area is the other half, as required.

Why Other Options Are Wrong:
1/2 would give area ratio 1/4. Other expressions are unrelated to the square-root relation.

Common Pitfalls:
Confusing linear and area ratios; forgetting that area scales with the square of length.

Final Answer:
1/√2