Equilateral triangle described on the diagonal of a square\nA square has side a. An equilateral triangle is constructed on the square’s diagonal as its side. What is the ratio of the area of the equilateral triangle to the area of the square?

Introduction / Context:
Here, the side of the equilateral triangle equals the diagonal of the square. Converting side relations into areas gives a clean ratio without needing the actual numeric value of a.

Given Data / Assumptions:

• Square side = a ⇒ diagonal d = a*√2
• Equilateral triangle side = d

Concept / Approach:
Area(square) = a^2. Area(equilateral) = (√3/4)*side^2. Substitute side = a√2.

Step-by-Step Solution:

Verification / Alternative check:
Let a = 2 for ease: diagonal = 2√2; triangle area = (√3/4)*(8) = 2√3; square area = 4 ⇒ ratio 2√3 : 4 = √3 : 2.

Why Other Options Are Wrong:
1:2, 1:3, and 2:3 do not arise from the exact formula for the equilateral triangle’s area based on the diagonal length.

Common Pitfalls:
Confusing the diagonal with the side of the square or forgetting to square the √2 factor when computing area of the triangle can derail the answer.

Final Answer:
√3:2