Square inscribed in a circle (radius 4 cm) with an external tangent:\nPQRS is a square inscribed in a circle of radius 4 cm (so the circle's diameter equals the square's diagonal). Side PQ is extended beyond Q to meet a point Y. From Y, a tangent is drawn to the circle touching it at point R. Find the exact length (in cm) of segment SY.

Introduction / Context:
This geometry problem mixes an inscribed square and a tangent from an external point. Converting the figure to coordinates makes the tangent-intersection computation straightforward and avoids guesswork.

Given Data / Assumptions:

• Circle center O with radius R = 4 cm.
• PQRS is a square inscribed in the circle (its diagonal equals the circle’s diameter).
• PQ is extended to a point Y; from Y, the tangent to the circle touches at R.
• Required: length SY.

Concept / Approach:
Place the circle at the origin. Choose the square aligned with axes, so vertices are at (±2√2, ±2√2). The tangent at a point of the circle is perpendicular to the radius to that point. Intersect the tangent at R with the horizontal line containing PQ to get Y, then compute SY by distance formula.

Step-by-Step Solution:

Verification / Alternative check:
Any square rotation about O keeps distances congruent. With symmetry, the computed value is invariant under rigid motions, confirming 4√10.

Why Other Options Are Wrong:
2√10 and 3√5 underestimate; 6√10 overestimates; 5√6 does not match the derived √160.

Common Pitfalls:
Using side as 4 (it is 4√2) or taking the tangent slope equal (not perpendicular) to the radius. Ensure tangent slope = −1 when radius slope = 1.

Final Answer:
4√10