In a right-angled triangle PQR with ∠Q = 90°, the side lengths are PQ = 12 cm and QR = 5 cm. Find the radius of the circumcircle (circumradius) of triangle PQR.

Introduction / Context:
This question tests a key geometry result: in a right triangle, the circumcenter lies at the midpoint of the hypotenuse, and the circumradius equals half the hypotenuse. So the task reduces to finding the hypotenuse using Pythagoras and then halving it.

Given Data / Assumptions:

• Triangle PQR is right-angled at Q (∠Q = 90°) • PQ = 12 cm, QR = 5 cm (legs) • PR is the hypotenuse • Circumradius R = PR/2 for a right triangle

Concept / Approach:
1) Use Pythagorean theorem: PR^2 = PQ^2 + QR^2. 2) Compute PR. 3) Use the right-triangle circumradius property: Circumradius = hypotenuse/2.

Step-by-Step Solution:
1) Apply Pythagoras: PR^2 = 12^2 + 5^2 2) Compute squares: 12^2 = 144, 5^2 = 25 3) Add: PR^2 = 144 + 25 = 169 4) Take square root: PR = √169 = 13 cm 5) Circumradius of a right triangle: R = PR/2 = 13/2 = 6.5 cm

Verification / Alternative check:
The circumcenter of a right triangle is the midpoint of the hypotenuse, so the radius must reach a hypotenuse endpoint. Distance from midpoint to endpoint is half the hypotenuse, confirming R = 6.5 cm.

Why Other Options Are Wrong:
• 13 cm is the hypotenuse, not the radius. • 6 cm, 5 cm, 7 cm: do not equal half of 13 and therefore cannot be the circumradius.

Common Pitfalls:
• Using (12 + 5)/2 instead of PR/2. • Forgetting that circumradius in a right triangle depends on the hypotenuse, not the legs directly.

Final Answer:
6.5 cm