A rectangle of dimensions 3 units by 4 units is inscribed in a circle. Assuming the rectangle’s vertices lie on the circle, find the circumference of the circle in terms of pi.

Introduction / Context: This problem tests the relationship between a rectangle and its circumcircle. When a rectangle is inscribed in a circle, the diagonal of the rectangle becomes the diameter of the circle. Using the Pythagoras theorem, the diagonal can be found easily.

Given Data / Assumptions:

• Rectangle dimensions = 3 units and 4 units
• Diagonal of rectangle = diameter of the circle
• Circumference of circle = pi * diameter

Concept / Approach: Use Pythagoras theorem to find the diagonal of the rectangle. Treat this diagonal as the diameter of the circle, then compute the circumference using the standard formula.

Step-by-Step Solution: Diagonal = sqrt(3^2 + 4^2) Diagonal = sqrt(9 + 16) = sqrt(25) = 5 Diameter of circle = 5 Circumference = pi * diameter = 5π

Verification / Alternative check: A 3–4–5 triangle is a well-known right triangle. Since the diagonal is exactly 5 units, the resulting circumference of 5π is consistent and exact.

Why Other Options Are Wrong: 2.5π and 3π: correspond to smaller diameters than the actual diagonal. 4π: assumes diameter equals the longer side, not the diagonal. 6π: overestimates the diagonal length.

Common Pitfalls: Using one side of the rectangle as the diameter. Forgetting that the diagonal is the diameter, not the radius. Arithmetic errors while applying Pythagoras theorem.

Final Answer: Circumference of the circle = 5π