A rectangle is inscribed in a circle. The lengths of its two adjacent sides are 5 cm and 12 cm. What is the radius of the circle (in centimetres)?

Introduction / Context:
This question tests your understanding of the relationship between a rectangle inscribed in a circle and the circle's diameter. It uses the Pythagorean theorem and the fact that the diagonal of a rectangle inscribed in a circle is always the diameter of that circle.

Given Data / Assumptions:

• A rectangle is drawn inside a circle so that all four vertices lie on the circle.
• The adjacent sides of the rectangle measure 5 cm and 12 cm.
• We need to find the radius of the circle.

Concept / Approach:
For any rectangle inscribed in a circle, the diagonal of the rectangle passes through the centre of the circle and is equal to the diameter of the circle. Therefore, if we find the length of the diagonal using the Pythagorean theorem, half of this diagonal will give the radius. In a rectangle with sides a and b, the diagonal d satisfies d^2 = a^2 + b^2. Here the sides are 5 cm and 12 cm, which form a well known Pythagorean triple.

Step-by-Step Solution:
Step 1: Let the sides of the rectangle be a = 5 cm and b = 12 cm. Step 2: Use the Pythagorean theorem to find the diagonal d: d^2 = a^2 + b^2. Step 3: Compute a^2 = 5^2 = 25 and b^2 = 12^2 = 144. Step 4: Add them: d^2 = 25 + 144 = 169. Step 5: Take the square root: d = √169 = 13 cm. Step 6: The diagonal is the diameter of the circle, so diameter = 13 cm. Step 7: Radius r = diameter / 2 = 13 / 2 = 6.5 cm.

Verification / Alternative Check:
You can recognise that 5, 12, 13 is a standard Pythagorean triple, so any right triangle or rectangle with legs 5 and 12 automatically has a hypotenuse or diagonal of 13. Since the rectangle's diagonal matches the circle's diameter, the radius must be half of 13, which is 6.5 cm. This quick recognition reinforces the calculation above.

Why Other Options Are Wrong:
A radius of 6 cm or 8 cm or 8.5 cm would correspond to diameters of 12 cm, 16 cm, or 17 cm respectively, none of which match the necessary 13 cm diagonal created by the 5–12 sides. This would contradict the geometry of a rectangle inscribed in a circle, where the diagonal must equal the diameter.

Common Pitfalls:
Sometimes learners confuse the radius with the diagonal itself and forget to divide by 2, wrongly choosing 13 instead of 6.5. Others might calculate the diagonal incorrectly by adding side lengths instead of adding the squares of the side lengths. Always remember that the Pythagorean theorem uses squared lengths, and that the radius is half the diameter, not the full diagonal.

Final Answer:
The radius of the circle is 6.5 cm.