In triangle ABC, which is right angled at A (∠BAC = 90°) and has ∠ACB = 60°, what is the ratio of the circumradius of the triangle to the side AB?

Introduction / Context:
This geometry question involves a right angled triangle and its circumradius. The triangle ABC is right angled at A and another angle is given as 60 degrees. We are asked to find the ratio of the circumradius to one specific side, AB. The problem tests knowledge of right triangle properties, 30-60-90 triangles and the relation between the hypotenuse and circumradius.

Given Data / Assumptions:
- Triangle ABC is right angled at A, so ∠BAC = 90°.
- ∠ACB = 60°, so the remaining angle ∠ABC is 30°.
- We need the ratio circumradius R : side AB.
- All sides are positive and the triangle is non-degenerate.

Concept / Approach:
For a right angled triangle, the circumcenter lies at the midpoint of the hypotenuse and the circumradius R is half of the hypotenuse. Also, a triangle with angles 30°, 60° and 90° has well known side ratios. By identifying which side corresponds to which angle, we can express R and AB in terms of a common scale factor and then take their ratio.

Step-by-Step Solution:
Step 1: Since ∠BAC = 90°, side BC is the hypotenuse.Step 2: ∠ACB = 60° means that side AB is opposite 60°, and ∠ABC = 30° means that side AC is opposite 30°.Step 3: In a standard 30°-60°-90° triangle, if the side opposite 30° is a, then the side opposite 60° is a√3 and the hypotenuse is 2a.Step 4: Matching this to our triangle, AC (opposite 30°) = a, AB (opposite 60°) = a√3, and BC (hypotenuse) = 2a.Step 5: For a right angled triangle, circumradius R = hypotenuse / 2 = BC / 2 = (2a) / 2 = a.Step 6: Therefore, R : AB = a : (a√3) = 1 : √3.

Verification / Alternative check:
You can assign a specific value to a, for example a = 1. Then AC = 1, AB = √3 and BC = 2. The circumradius is BC / 2 = 1. So the ratio R : AB = 1 : √3, which confirms our symbolic result.

Why Other Options Are Wrong:
The ratio 1 : 2 would correspond to R being half of AB, which is incorrect because the hypotenuse, not AB, is directly related to R. Ratios like 2 : √3 or 2 : 3 come from confusing which side corresponds to which angle or mixing up the standard 30°-60°-90° side ratios.

Common Pitfalls:
Students sometimes misidentify the hypotenuse or pair the wrong side with the wrong angle, which leads to wrong ratios. Another frequent mistake is assuming R is half of any side instead of specifically half of the hypotenuse in a right angled triangle. Remember always to map each angle to its opposite side before applying known ratios.

Final Answer:
The required ratio of circumradius to side AB is 1 : √3.