A triangle has side lengths 9 cm, 12 cm, and 15 cm. Find the radius of its circumcircle (the circle passing through all three vertices).

Introduction / Context:
This question asks for the circumradius of a triangle whose side lengths are known. In particular, the triangle with sides 9 cm, 12 cm, and 15 cm forms a right triangle because it is a multiple of the 3, 4, 5 Pythagorean triplet. Right triangles have a simple relation between the hypotenuse and the circumradius, making this question a useful test of both recognition of special triangles and knowledge of circle geometry.

Given Data / Assumptions:

• Side lengths of the triangle are 9 cm, 12 cm, and 15 cm.
• The triangle is non degenerate and lies in a plane.
• We are asked to find the circumradius R of the triangle.
• For a right triangle, the hypotenuse is the diameter of the circumcircle.
• All units are in centimetres.

Concept / Approach:
First, we check whether the triangle is right angled by using the Pythagoras theorem. If the square of the longest side equals the sum of the squares of the other two sides, then the triangle is right angled, and the longest side is the hypotenuse. For a right triangle, the circumcircle has diameter equal to the hypotenuse, so the circumradius R is hypotenuse / 2. Once this is confirmed, we simply divide the hypotenuse length by 2 to obtain R.

Step-by-Step Solution:
Identify the longest side: 15 cm.Check Pythagoras: 9^2 + 12^2 = 81 + 144 = 225.Compute 15^2 = 225, so 9^2 + 12^2 = 15^2, confirming a right triangle with hypotenuse 15 cm.For a right triangle, circumdiameter = hypotenuse, so diameter D = 15 cm.Circumradius R = D / 2 = 15 / 2 = 7.5 cm.

Verification / Alternative check:
Another way to compute the circumradius of any triangle is to use the formula R = (a * b * c) / (4 * Δ), where a, b, and c are side lengths and Δ is the area. For a right triangle with legs 9 and 12, area Δ = (1 / 2) * 9 * 12 = 54. Substituting into the formula gives R = (9 * 12 * 15) / (4 * 54) = 1620 / 216 = 7.5 cm. This matches the simpler hypotenuse based method, confirming the result.

Why Other Options Are Wrong:
Option 6 cm would give a diameter of 12 cm, which is shorter than the hypotenuse, impossible for a right triangle circumcircle. Option 4.5 cm is even smaller and does not satisfy the formula for R. Option 8 cm gives diameter 16 cm, which does not match any relation with the triangle sides. Option 9.5 cm is arbitrarily larger than needed and fails when substituted into standard circumradius formulas.

Common Pitfalls:
Some students fail to recognize the 9, 12, 15 triangle as a scaled 3, 4, 5 triangle and attempt more complicated methods. Others misidentify which side is the hypotenuse or apply area formulas incorrectly. Forgetting the special property that the hypotenuse is the diameter of the circumcircle for right triangles is another common source of mistakes.

Final Answer:
The circumradius of the triangle is 7.5 cm.