An equilateral triangle is constructed on the diagonal of a square (the diagonal is the side of the equilateral triangle). What is the ratio of the area of the triangle to the area of the square?

Introduction / Context:This question tests linking two shapes using a shared measurement and then comparing their areas. The diagonal of the square becomes the side of the equilateral triangle. For a square with side s, diagonal is s*sqrt(2). For an equilateral triangle with side a, area is (sqrt(3)/4)*a^2. Square area is s^2. The ratio of areas is found by expressing the triangle side a in terms of s (using the diagonal relationship), substituting into the triangle area formula, and then dividing by s^2. The result is a clean symbolic ratio involving sqrt(3).

Given Data / Assumptions:

• Square side = s
• Square diagonal = s*sqrt(2)
• Equilateral triangle side a = square diagonal
• Area(square) = s^2
• Area(equilateral triangle) = (sqrt(3)/4)*a^2

Concept / Approach:Let a = s*sqrt(2). Then triangle area = (sqrt(3)/4)*(s^2*2) = (sqrt(3)/2)*s^2. Ratio (triangle:square) = (sqrt(3)/2)*s^2 : s^2 = sqrt(3):2.

Step-by-Step Solution: Let square side = s Diagonal = s*sqrt(2) Triangle side a = s*sqrt(2) Area(triangle) = (sqrt(3)/4)*a^2 a^2 = (s*sqrt(2))^2 = 2*s^2 Area(triangle) = (sqrt(3)/4)*2*s^2 = (sqrt(3)/2)*s^2 Area(square) = s^2 Ratio = (sqrt(3)/2)*s^2 : s^2 = sqrt(3):2

Verification / Alternative check:If s=1, square area is 1. Diagonal is sqrt(2), triangle area becomes (sqrt(3)/4)*2 = sqrt(3)/2. Ratio is sqrt(3)/2 : 1 which corresponds to sqrt(3):2. This confirms the symbolic ratio is consistent.

Why Other Options Are Wrong: 1:2, 1:3, 2:3 are plain fractions and ignore the sqrt(3) factor from equilateral triangle area. 3:2 incorrectly replaces sqrt(3) with 3.

Common Pitfalls:Using triangle area as (1/2)*base*height without correct height, forgetting diagonal = s*sqrt(2), or squaring sqrt(2) incorrectly.

Final Answer:The ratio of the area of the triangle to the area of the square is sqrt3:2.