The diagonal of a square is 20 cm. Using the relationship between a square's side and its diagonal, find the perimeter of the square (give the exact form and a useful decimal approximation).

Introduction / Context:
This question tests the geometry of a square. A square's diagonal is related to its side by the Pythagoras theorem because the diagonal is the hypotenuse of a right triangle whose legs are the equal sides of the square. Once the side is found, perimeter is simply 4 times the side.

Given Data / Assumptions:

• Diagonal, d = 20 cm
• Let side length be s cm
• In a square: d^2 = s^2 + s^2 = 2s^2
• Perimeter = 4s

Concept / Approach:
Use d = s*sqrt(2) or equivalently 2s^2 = d^2. Solve for s and then compute perimeter = 4s. Provide both exact and approximate value for clarity.

Step-by-Step Solution:
d^2 = 2s^2 20^2 = 2s^2 400 = 2s^2 s^2 = 200 s = sqrt(200) = 10*sqrt(2) cm (since sqrt(200) = sqrt(100*2) = 10*sqrt(2)) Perimeter = 4s = 4*(10*sqrt(2)) = 40*sqrt(2) cm Approximation: sqrt(2) ≈ 1.414, so perimeter ≈ 40*1.414 = 56.57 cm

Verification / Alternative check:
If s = 10*sqrt(2), then diagonal d should be s*sqrt(2) = 10*sqrt(2)*sqrt(2) = 10*2 = 20 cm, matching the given value.

Why Other Options Are Wrong:
40 cm: that would be perimeter only if side were 10 cm, but then diagonal becomes 10*sqrt(2) ≈ 14.14 cm, not 20 cm. 20*sqrt(2) cm: that is actually the diagonal for side 10 cm, not the perimeter. 80 cm: would imply side 20 cm, making diagonal 20*sqrt(2) ≈ 28.28 cm. 56 cm: rounding too aggressively and not consistent with 56.57 cm; the intended precise result is 40*sqrt(2).

Common Pitfalls:
Confusing the diagonal formula d = s*sqrt(2) with perimeter. Using d = 2s (which is incorrect for a square). Forgetting to multiply the side by 4 to get perimeter.

Final Answer:
Perimeter = 40*sqrt(2) cm (approximately 56.57 cm)