**Core Concept**
The anatomical snuff box is a triangular deepening on the radial, dorsal aspect of the hand. It is bounded medially by the tendon of the extensor pollicis longus and laterally by the tendon of the extensor pollicis brevis. The radial border of the anatomical snuff box is typically associated with the tendon of the abductor pollicis longus.

**Why the Correct Answer is Right**
The abductor pollicis longus muscle is responsible for abducting the thumb radially. The tendon of this muscle is closely associated with the radial border of the anatomical snuff box. This tendon can be palpated at the radial border of the anatomical snuff box, making it a useful landmark in clinical examination. The tendon's location is due to its anatomical course, which runs along the radial border of the wrist.

**Why Each Wrong Option is Incorrect**
* **Option A:** The extensor pollicis brevis muscle is located at the lateral border of the anatomical snuff box, not the radial border.
* **Option B:** The extensor pollicis longus muscle is located at the medial border of the anatomical snuff box, not the radial border.
* **Option D:** The extensor carpi radialis brevis tendon is located more proximally on the lateral aspect of the forearm, not at the radial border of the anatomical snuff box.

**Clinical Pearl / High-Yield Fact**
The anatomical snuff box is a crucial landmark in clinical examination, particularly for palpating the tendons of the thumb extensors. The radial border of the anatomical snuff box is associated with the tendon of the abductor pollicis longus, which can be a useful landmark for diagnosing conditions affecting the thumb.

**Correct Answer:** C.