Cyclic quadrilateral with equal opposite angles (rectangle case): A quadrilateral inscribed in a circle has equal opposite angles (hence it is a rectangle). If its adjacent sides are 6 cm and 8 cm, find the area of the circumcircle.

Introduction / Context:In a cyclic quadrilateral, equal opposite angles imply each is 90°, i.e., the figure is a rectangle. The circle circumscribing a rectangle has diameter equal to the rectangle’s diagonal.

Given Data / Assumptions:

• Rectangle sides: 6 cm and 8 cm.
• Diagonal d = √(6^2 + 8^2) = 10 cm.
• Circumcircle diameter = diagonal; hence radius r = d/2 = 5 cm.

Concept / Approach:Area of the circle = πr^2 = π * 25 = 25π sq cm.

Step-by-Step Solution:

Verification / Alternative check:The rectangle’s vertices lie on a circle with diameter equal to the diagonal; this is a standard property of right angles in semicircles.

Why Other Options Are Wrong:64π, 36π, 49π, 50π do not equal π * 5^2.

Common Pitfalls:Taking side as diameter or forgetting that the rectangle’s circumcircle radius depends on the diagonal, not any single side.

Final Answer:25π sq cm