**Core Concept**
The girl's symptoms suggest a neuromuscular disorder, specifically Duchenne Muscular Dystrophy (DMD), characterized by progressive muscle weakness, particularly affecting proximal muscles, and a distinctive clinical presentation.

**Why the Correct Answer is Right**
The positive Gower's sign, which involves the patient using their hands to "walk" up their legs to stand up from a lying position, is a classic sign of DMD. This occurs due to the gradual degeneration of muscle fibers, particularly in the lower limbs, leading to significant muscle weakness. The maculopapular rash over the metacarpophalangeal joints may be related to Becker's rash, a characteristic feature of DMD. The next appropriate investigation would be to confirm the diagnosis by detecting the absence or mutation of the dystrophin gene, which is responsible for the disease.

**Why Each Wrong Option is Incorrect**
* **Option A:** Creatine kinase (CK) levels would be elevated in DMD but are not the next investigation to confirm the diagnosis.
* **Option B:** Electromyography (EMG) can demonstrate muscle denervation but is not the primary investigation for confirming DMD.
* **Option C:** Muscle biopsy can demonstrate dystrophin deficiency but is not the first-line investigation for DMD diagnosis.

**Clinical Pearl / High-Yield Fact**
Becker's rash, a maculopapular rash over the metacarpophalangeal joints, can be a distinguishing feature of Duchenne Muscular Dystrophy (DMD) from other muscular dystrophies.

**Correct Answer:** D. Muscle biopsy