Find the ratio of the area of a square to the area of the square drawn on its diagonal (i.e., the diagonal of the first square becomes the side of the second square).

Introduction:
This question tests how a square’s diagonal relates to its side and how that affects area. The diagonal of a square is longer than its side by a factor of sqrt(2). If a new square is constructed on the diagonal of the original square, that diagonal becomes the side length of the new square. Since area depends on side^2, the ratio of areas becomes a clean constant. The key is to express both areas in terms of the original side length s and simplify.

Given Data / Assumptions:

• Let original square side = s• Original square area = s^2• Original diagonal = s*sqrt(2)• New square is drawn on the diagonal, so its side = s*sqrt(2)

Concept / Approach:
Diagonal of a square: d = s*sqrt(2). New square side equals d, so new area = d^2 = (s*sqrt(2))^2 = 2s^2. Ratio original:new = s^2 : 2s^2 = 1:2.

Step-by-Step Solution:
Step 1: Assume original side = s.Original area = s^2Step 2: Compute diagonal of original square.Diagonal d = s*sqrt(2)Step 3: New square has side equal to the diagonal.New side = d = s*sqrt(2)Step 4: Compute new square area.New area = (s*sqrt(2))^2 = s^2 * 2 = 2s^2Step 5: Form ratio of areas.Original : New = s^2 : 2s^2 = 1 : 2

Verification / Alternative check:
Let s = 10. Original area = 100. Diagonal = 10*1.414 ≈ 14.14. New square area = (14.14)^2 ≈ 200. Ratio 100:200 = 1:2. The numerical check confirms the algebraic ratio exactly.

Why Other Options Are Wrong:
2:1 would imply the new square has smaller area, which is impossible because its side is the diagonal (longer than s).2:3, 3:1, and 4:1 are inconsistent with the exact factor of 2 produced by squaring sqrt(2).Only 1:2 matches the exact relationship.

Common Pitfalls:
• Forgetting that diagonal = s*sqrt(2) (using s/2 or another incorrect relation).• Comparing side lengths instead of areas (areas require squaring).• Reversing the ratio while writing the final answer.

Final Answer:
The ratio of areas is 1 : 2.