What is the most appropriate investigation for a 60-year-old man with symptoms of thirst, urinary frequency, and weight loss?

The symptoms of thirst, urinary frequency, and weight loss are classic indicators of hyperglycemia, often associated with diabetes mellitus. In adults, particularly those who are 60 years old, these symptoms could signal the onset of type 2 diabetes, characterized by insulin resistance and insufficient insulin secretion.

The most appropriate investigation for confirming this condition is to measure blood glucose levels. A randomized blood sugar test or a fasting plasma glucose test can help determine if the patient has elevated glucose levels, which would support a diagnosis of diabetes. Additionally, an HbA1c test may also be utilized to assess long-term glycemic control, but the immediate step would be to check the blood sugar.

Other investigations listed are not relevant in this context. Arteriography and venography are methods used to image blood vessels and are unrelated to diagnosing metabolic disorders like diabetes. A biopsy for malignant melanoma is specific to certain skin conditions and does not address the symptoms presented. Therefore, measuring blood sugar is the critical step in this clinical scenario.