Incised square — A circle has circumference 25 cm. Find the side of a square inscribed inside it.

Introduction / Context:For a square inscribed in a circle, the square’s diagonal equals the circle’s diameter. If we know the circumference, we can find the radius/diameter and hence the square’s side.

Given Data / Assumptions:

• Circumference C = 25 cm
• 2πr = 25 ⇒ r = 25/(2π)
• Diameter D = 2r = 25/π
• Inscribed square side s = D/√2 = (25/π)/√2

Concept / Approach:Use the relation between an inscribed square and its circumcircle (the given circle). Side is diagonal divided by √2.

Step-by-Step Solution:s = (25/π)/√2 = 25/(π√2) cm

Verification / Alternative check:Compute the square’s diagonal s√2 = 25/π, matching the circle’s diameter as required.

Why Other Options Are Wrong:Other expressions use wrong radicals or constants; only 25/(π√2) follows from geometry.

Common Pitfalls:Using the radius instead of the diameter for the square’s diagonal; missing the √2 relationship.

Final Answer:25/(π√2) cm