Squares and their diagonals:\nFind the ratio of the area of a square to the area of the square built on its diagonal.

Introduction / Context:
Building a new square on the diagonal of an original square scales the side by √2, so area doubles.

Given Data / Assumptions:
Original square side = s; its diagonal = s√2, which becomes the side of the new square.

Concept / Approach:
Areas: original = s^2; new = (s√2)^2 = 2s^2. Ratio original:new = 1:2.

Step-by-Step Solution:

Verification / Alternative check:
Let s = 10 → original 100; new 200 ⇒ 1:2.

Why Other Options Are Wrong:
2:3, 3:1, 4:1 do not match the √2 side scaling effect on area.

Common Pitfalls:
Using √2 for ratio directly; remember area scales with square of side.

Final Answer:
1:2