**Core Concept**
Hemiplegia is a neurological condition characterized by weakness or paralysis of one side of the body, typically resulting from damage to the brain's motor control pathways. The most common cause of hemiplegia is vascular in nature, specifically involving the cerebral vasculature.

**Why the Correct Answer is Right**
Thrombosis of the middle cerebral artery (MCA) is the most common cause of hemiplegia due to its large territory of supply and the significant motor cortex it serves. The MCA is responsible for controlling the motor functions of the face, arm, and leg on the contralateral side of the body. Thrombosis of the MCA leads to ischemia and infarction of the affected motor cortex, resulting in hemiplegia. The MCA's supply area includes the motor homunculus, a somatotopic representation of the body's motor functions in the brain.

**Why Each Wrong Option is Incorrect**
**Option A:** Thrombosis of the anterior cerebral artery (ACA) is less likely to cause hemiplegia, as it primarily supplies the medial surface of the brain and is involved in controlling lower limb motor functions.

**Option B:** Thrombosis of the posterior cerebral artery (PCA) is more likely to cause visual disturbances and less likely to cause hemiplegia, as it primarily supplies the occipital lobe.

**Option C:** Thrombosis of the vertebral artery or basilar artery is more likely to cause brainstem or cerebellar dysfunction, rather than hemiplegia.

**Clinical Pearl / High-Yield Fact**
The "MCA" is a high-yield fact for exam purposes, as it is the most common cause of hemiplegia. Remember the acronym "MCA" to stand for "Motor Cortex Affected" in the context of stroke.

**Correct Answer: C. Thrombosis of the vertebral artery or basilar artery.**