A quadrilateral is inscribed in a circle. Its opposite angles are equal, and the lengths of two adjacent sides are 6 cm and 8 cm. What is the area of the circle (in terms of π)?

Introduction / Context:
This problem involves a quadrilateral inscribed in a circle, also called a cyclic quadrilateral. The condition that opposite angles are equal has a strong implication for the type of quadrilateral, and knowing this type allows us to infer more about the geometry and the circle. The question asks for the area of the circumscribed circle using the lengths of two adjacent sides of the quadrilateral.

Given Data / Assumptions:

• A quadrilateral is inscribed in a circle, so all its vertices lie on the circle.
• Opposite angles of the quadrilateral are equal.
• Two adjacent sides of the quadrilateral measure 6 cm and 8 cm.
• We are asked to find the area of the circle in terms of π.
• All lengths are in centimetres.

Concept / Approach:
In any cyclic quadrilateral, the sum of a pair of opposite angles is 180°. If opposite angles are also equal, then each must be 90°, making the quadrilateral a rectangle. In a rectangle with sides 6 cm and 8 cm, the diagonal can be found using Pythagoras theorem. That diagonal is also the diameter of the circle containing the rectangle. Once the diameter is known, the radius is half the diameter, and the area of the circle is π * r^2.

Step-by-Step Solution:
Opposite angles equal and supplementary (sum 180°) imply each opposite angle is 90°, so the quadrilateral is a rectangle.Let the rectangle sides be 6 cm and 8 cm.Diagonal of the rectangle (and diameter of the circle) is d = √(6^2 + 8^2) = √(36 + 64) = √100 = 10 cm.Radius r of the circle is r = d / 2 = 10 / 2 = 5 cm.Area of the circle = π * r^2 = π * 5^2 = 25π square centimetres.

Verification / Alternative check:
If we draw a rectangle of sides 6 and 8, placing it inside a circle so that its corners touch the circle, the diagonal clearly spans the circle. This is equivalent to the well known 6, 8, 10 right triangle forming half of the rectangle. Since the diagonal length is 10, the radius of the circle must be 5. There is no alternative circle with a different radius that can pass through all four vertices of this rectangle, confirming the uniqueness and correctness of the area 25π.

Why Other Options Are Wrong:
Option 64π would correspond to a radius of 8, which would make the diagonal 16, not matching the rectangle dimensions. Option 36π corresponds to a radius of 6, giving a diameter of 12 instead of 10. Option 49π is based on a radius of 7, again not compatible with a 6 by 8 rectangle. Option 81π would require a radius of 9, which is far too large for the given side lengths.

Common Pitfalls:
Some students may not recognize that equal opposite angles in a cyclic quadrilateral force each to be 90°, leading to a rectangle. Others may mistakenly think any quadrilateral with equal opposite angles is a general parallelogram and forget the extra condition of being cyclic. Miscomputing the diagonal or confusing diameter with radius are further sources of errors.

Final Answer:
The area of the circle is 25π sq cm.