In geometry, the circumradius of a right-angled triangle equals half the length of its hypotenuse.\nTriangle XYZ is right-angled at Y (angle Y = 90 degrees). If XY = 2.5 cm and YZ = 6 cm, what is the circumradius of triangle XYZ (in cm)?

Introduction / Context:
This question checks a key circle-geometry fact: for a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse, so the circumradius is exactly half of the hypotenuse. The only computation needed is the hypotenuse length using Pythagoras.

Given Data / Assumptions:

• Triangle XYZ is right-angled at Y (angle Y = 90 degrees).
• Legs: XY = 2.5 cm and YZ = 6 cm.
• Hypotenuse: XZ.
• Circumradius R = (hypotenuse)/2 for a right triangle.

Concept / Approach:
First find the hypotenuse using Pythagoras: XZ = sqrt(XY^2 + YZ^2). Then apply the right-triangle circumradius property: R = XZ/2. This property is a standard result because the diameter of the circumcircle of a right triangle is the hypotenuse (Thales' theorem).

Step-by-Step Solution:
Compute the hypotenuse: XZ^2 = XY^2 + YZ^2 XZ^2 = (2.5)^2 + (6)^2 = 6.25 + 36 = 42.25 XZ = sqrt(42.25) = 6.5 cm For a right triangle, circumradius R = XZ / 2 R = 6.5 / 2 = 3.25 cm

Verification / Alternative check:
If the circumcircle diameter is the hypotenuse, then diameter = 6.5 cm and radius must be 3.25 cm. This directly matches the computed value.

Why Other Options Are Wrong:
6.5 cm is the hypotenuse itself (diameter), not the radius.
4 cm and 3 cm come from incorrect halving or rounding of the hypotenuse.
2.5 cm incorrectly uses one leg as the radius.

Common Pitfalls:
Forgetting that the special circumradius rule applies only when the triangle is right-angled, or halving the wrong side (a leg instead of the hypotenuse). Another mistake is squaring incorrectly when computing the hypotenuse.

Final Answer:
The circumradius is 3.25 cm